Answer:
[tex]\large\boxed{x=-2\ or\ x=-\dfrac{1}{3}}[/tex]
Step-by-step explanation:
The quadratic formula of a quadratic equation
[tex]ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
We have the equation:
[tex]3p^2+7p+2=0\to a=3,\ b=7,\ c=2[/tex]
Substitute:
[tex]b^2-4ac=7^2-4(3)(2)=49-24=25\\\\\sqrt{b^2-4ac}=\sqrt{25}=5\\\\x_1=\dfrac{-7-5}{2(3)}=\dfrac{-12}{6}=-2\\\\x_2=\dfrac{-7+5}{2(3)}=\dfrac{-2}{6}=-\dfrac{1}{3}[/tex]
Answer:
-1/3,-2
Step-by-step explanation:
The quadratic formula is [-b±√(b^2-4ac)]/2a. In this equations 3p^2 is a, 7p is b, and 2 is c.
Then just plug in the numbers.
[-7±√((-7^2)-4(3)(2)]/2(3)
[-7±√(25)]/6
(-7+5)/6 and (-7-5)/6
-1/3 and -2 are the answers, if you plug these numbers into the original equation, you find that they equal 0 which means that they work.
-3(8/9)÷66.9[9(-8.9+9/5).01]
i neeeed help or il fail the 6weeks
The answer of -3(8/9) ÷ 66.9[9 (-8.9 + 9/5) .01] is 0.06237948089
Step-by-step explanation:
To solve this problem -3(8/9) ÷ 66.9[9 (-8.9 + 9/5) .01] we must start with
1. Solve this bracket [9 (-8.9 + 9/5) .01] at first
2. Multiply the answer of the bracket by 66.9
3. Multiply -3(8/9) and divide the answer by the product of the previous
step
In [9 (-8.9 + 9/5) .01]
∵ (-8.9 + 9/5) = (-8.9 + 1.8) = -7.1
∴ [9 (-8.9 + 9/5) .01] = [9 (-7.1).01]
∵ 9(-7.1)(.01) = -0.639
∴ [9 (-8.9 + 9/5) .01] = -0.639
∵ 66.9 [-0.639] = -42.7491
∵ [tex]-3(\frac{8}{9})=\frac{-8}{3}[/tex]
∴ -3(8/9) ÷ 66.9[9 (-8.9 + 9/5) .01] = [tex]\frac{-8}{3}[/tex] ÷ -42.7491
∴ -3(8/9) ÷ 66.9[9 (-8.9 + 9/5) .01] = 0.06237948089
The answer of -3(8/9) ÷ 66.9[9 (-8.9 + 9/5) .01] is 0.06237948089
Learn more:
You can learn more about decimals in brainly.com/question/1774450
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How do you convert 2/11 into a decimal ?
Answer:
you would divide the two numbers. 2 divided by 11 is 0.18
Answer:
0.181818182 or rounded is 0.18
Step-by-step explanation:
to turn any fraction into a decimal you always do numerator divided by denominator
the Scientific notation 2 times 10^10-2 has what value
2 x 10^(10 - 2)
2 x 10^(8)
2 x 100,000,000
200, 000, 000
One can use two-dimensional objects to build three-dimensional objects?
True or False?
Answer:
TRUE
Step-by-step explanation:
NEED HELP ASAP!!!
Given:
a | | b, c | | d
Prove:
m1 = m16
*Look at graphs attached*
Which of the following would be the reasons for statements 3 and 4 in the proof?
A. Vertical angles are equal.
B. If lines are ||, corresponding angles are equal.
C. If lines are ||, alternate interior angles are equal.
D. If lines are ||, alternate exterior angles are equal.
Answer:
3 and 4: B
Step-by-step explanation:
As per the property of traversal lines, when two parallel lines are cut by traversal then the corresponding angles formed are equal.
In given case ∠6 is corresponding angle of ∠8 and ∠8 is corresponding to ∠18.
Hence option B is correct for 3 and statements!
If lines are ||, corresponding angles are equal.
Answer:
If lines are ||, corresponding angles are equal.
Step-by-step explanation:
yep
Use the graph of the line to answer the questions.
What is an equation of the line in point-slope form?
How can the point-slope form be written in function notation?
Answer:
[tex]y+1=\dfrac{1}{3}(x+2)[/tex] - point-slope form
[tex]f(x)=\dfrac{1}{3}x-\dfrac{1}{3}[/tex] - function notation
Step-by-step explanation:
The point-slope form of an equation of a line:
[tex]y-y_1=m(x-x_1)[/tex]
m - slope
The formula of a slope:
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]
From the graph we have the points (-2, -1) and (1, 0).
Substitute:
[tex]m=\dfrac{0-(-1)}{1-(-2)}=\dfrac{1}{3}[/tex]
[tex]y-(-1)=\dfrac{1}{3}(x-(-2))[/tex]
[tex]y+1=\dfrac{1}{3}(x+2)[/tex] - point-slope form
[tex]y+1=\dfrac{1}{3}(x+2)[/tex] use the distributive property
[tex]y+1=\dfrac{1}{3}x+\dfrac{2}{3}[/tex] subtract 1 = 3/3 from both sides
[tex]y=\dfrac{1}{3}x-\dfrac{1}{3}[/tex]
Answer:
Point slope form : [tex]y-0=\frac{1}{3}(x-1)[/tex]
Function notation : [tex]f(x)=\frac{1}{3}(x-1)[/tex]
Step-by-step explanation:
If a line passes through two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the point slope form of line is
[tex]y-y_1=m(x-x_1)[/tex]
Where, m is slope.
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]
From the given graph it is clear that he line passes through the points (-2,-1) and (1,0). Slope of the line is
[tex]m=\frac{0-(-1)}{1-(-2)}=\frac{1}{3}[/tex]
The point is (1,0) and slope is 1/3. So, the point slope form of the line is
[tex]y-0=\frac{1}{3}(x-1)[/tex]
[tex]y=\frac{1}{3}(x-1)[/tex]
Therefore the point slope form is [tex]y-0=\frac{1}{3}(x-1)[/tex].
Replace y by f(x) to write the equation in function notation.
[tex]f(x)=\frac{1}{3}(x-1)[/tex]
Therefore the function notation is [tex]f(x)=\frac{1}{3}(x-1)[/tex].
Simplify negative 5 minus the square root of negative 44
Answer:
Simplification of [tex]-5-\sqrt{-44}[/tex] is:
[tex]-5-2i\sqrt{11}[/tex]
Step-by-step explanation:
We are given a expression:
[tex]-5-\sqrt{-44}[/tex]
We have to simplify the given expression.
We know that:
[tex]\sqrt{-1}=i[/tex]
[tex]-5-\sqrt{-44}\\\\=-5-\sqrt{-4\times 11}\\ \\=-5-\sqrt{-1\times 2^2\times 11}\\ \\=-5-2i\sqrt{11}[/tex]
Hence, simplification of [tex]-5-\sqrt{-44}[/tex] is:
[tex]-5-2i\sqrt{11}[/tex]
Select the correct answer from the drop-down menu.
Consider the equations y=√x and y=x² -1
The system of equations is equal at approximately
A.(1.5,1.2)
B.(-1.5,-1.2)
C.(1.5,-1.2)
D.(-1.5,1.2)
Answer:
Choice A.
Step.-by-step explanation:
y=√x
y=x² -1
Substituting the values in choice A:
1.2 = √1.5 = 1.2247 Approximately equal.
1.2 = (1.5)^2 - 1 = 1.25 Approximately equal.
Choice B.
-1.2 = √-1.5 which is imaginary so NOT this one.
Choice C
-1.2 = √1.5 = -1.2247
-1.2 = (1.5)^2 - 1 = 1.25 NO.
Choice D
We have the non real value √-1.5 again so NOT this one.
Answer: Option A
A.(1.5,1.2)
Step-by-step explanation:
We have the following system of equations
[tex]y=\sqrt{x}[/tex]
[tex]y=x^2 -1[/tex]
the solution of the system will be all points that satisfy both equations at the same time
For (1.5,1.2)
[tex]y=\sqrt{1.5}=1.2[/tex]
[tex]y=(1.5)^2 -1=1.2[/tex]
Both equations are satisfied
Note that we can discard options B and D because the domain of the equation [tex]y =\sqrt{x}[/tex] does not include the negative numbers.
We can discard option C because the range of the function [tex]y =\sqrt{x}[/tex] does not include the negative numbers.
Finally the answer is the option A
Which of the diagrams below represents the statement "If it is an triangle,
then it has three vertices"?
Answer:
I believe it's A, because all triangles have 3 vertices.
Step-by-step explanation:
Answer:
Figure A
Step-by-step explanation:
We are given that a statement
''If is is an triangle , then it has three vertices.
We have to find that which diagram represents the given statement
From first diagram
If triangle then it have three vertices because intersection of triangle and three vertices is triangle.
From second diagram
Intersection region of three vertices and triangle is three vertices.
So, from second diagram we cannot say if a triangle then it has three vertices.
Hence, figure A represent the given statement.
Angela has a marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of a?
Answer:
2
Step-by-step explanation:
Angela has a marbles.
Brian has twice as many as Angela, so Brain has 2a.
Caden has three times as many as Brain, so Caden has 3(2a)=6a.
Daryl has five times the number of marbles as Caden, so Daryl has 5(6a)=30a.
We are given that these 4 people together have 78 marbles.
Angela's + Brain's + Caden's + Daryl's =78
a + 2a + 6a + 30a =78
Combine the like terms:
39a=78
Divide both sides by 39:
a=78/39
a=2
Use the inverse relationship to complete the expression.
If i =
-1, then i^2=
Answer:
-1 edge
Step-by-step explanation:
...
1st-1
2nd –i
I just did it on edge :D
A mouse traveled a total distance of 3/24 of a mile in a maze over the past three hours the mouse travel the same distance each hour to determine the distance that the mouse traveled age our map reformed the calculations below he concluded that the mouse travel 3/8 of a mile each hour what is Matt’s error
Answer:
Total distance mouse traveled in 3 hours = [tex]\frac{3}{24}[/tex] of a mile
The mouse traveled the same distance in each hour. So in order to find the distance covered in 1 hour we have to divide the distance covered in 3 hours by 3. This will give us the distance that the mouse traveled in one hour.
So, the distance traveled in one hour will be = [tex]\frac{3}{24} \div 3 = \frac{3}{24} \times \frac{1}{3} =\frac{1}{24}[/tex] of a mile
The error which Matt made was that he divided only the denominator of the expression by 3, this probably was a calculation error.
Correct conclusion will be: Mouse travel 1/24 of a mile each hour
Answer:
Matt should have divided 3 by 3, not 24 by 3. (C.)
Step-by-step explanation:
since its a whole number you do not divide it at all.
Also:
Saloman was told the written paper on his science experiment should express measurements in liters.
Units of Capacity
Metric System Units Customary System Units
1 liter
0.26 gallon
1 liter
1.05 quarts
1 liter
4.22 cups
Which shows how to calculate the number of liters he used if he used 5 cups of water?
divide 5 by 4.22
multiply 5 by 4.22
divide 5 by 1.05
multiply 5 by 1.05
Final answer:
To find the liters for 5 cups of water, divide 5 by 4.22, according to the conversion rate obtained from the given table.
Explanation:
To calculate the number of liters used if Saloman used 5 cups of water, we need to determine how many cups are in a liter. According to the unit conversion provided, 1 liter is equivalent to approximately 4.22 cups. To find the number of liters from the number of cups, we should divide the total cups by the number of cups per liter. Therefore, to find the liters for 5 cups, we divide 5 by 4.22.
The calculation is: 5 cups ÷ 4.22 cups/liter = number of liters.
How many dimensions does a plane have?
O A. One
O B. Three
O C. Two
O D. Zero
Answer is C. Two
Answer:
C: Two dimensions
Step-by-step explanation:
Yes: a plane has two dimensions; think of a plane as being defined by the x- and y-axes, which intersect at a point in the plane.
A plane have two dimensions.
What is plane?A plane is a flat, two-dimensional surface that prolongs infinitely far. A plane is a two-dimensional analogue that could consist of a point, a line and three-dimensional space.
A plane is a flat, two-dimensional surface that extends indefinitely. It has two dimensions because every point in the plane can be described by a linear combination of two independent vectors.
Hence, a plane have two dimensions.
Find out more information about plane here:
https://brainly.com/question/17515776
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(3x^2-5x-7x^4)-(-2x^3+6x^4-5x^2)
Answer:
-13x^4+2x^3+8x^2-5x
Step-by-step explanation:
(3x^2-5x-7x^4)-(-2x^3+6x^4-5x^2)
= 3x^2-5x-7x^4+2x^3-6x^4+5x^2
= -13x^4+2x^3+8x^2-5x
Answer:
-x(13x³ - 2x² - 8x + 5)
Step-by-step explanation:
1. Distribute the negative to the numbers inside the parenthesis. Remember that two negatives multiplied will make a positive, and a negative and a positive will become a negative.
3x² - 5x - 7x^4 + 2x³ - 6x^4 + 5x²
2. Combine like terms. Remember that only numbers with the same number and exponents can be combined.
3x² - 5x - 7x^4 + 2x³ - 6x^4 + 5x²
↓
8x² - 5x - 7x^4 + 2x³ - 6x^4
↓
8x² - 5x - 13x^4 + 2x³
3. Rewrite the answer in descending order of powers.
-13x^4 + 2x³ + 8x² - 5x
4. Simplify The equation. The terms all have x in common, so we will take that out first. The smallest amount of x's any term has is one (-5x only has one x), so the most we can take out is one x. We will do this by lowering the power of every x by one.
x(-13x³ + 2x² + 8x - 5)
5. Since 13 and 5 are prime and do not go into 2 or 8 we will not be simplifying the coefficients. However, the last thing we can do is take out a negative.
-x(13x³ - 2x² - 8x + 5)
What are the zeros of f(x) = x2 - 12x + 36?
O
A. x= -6 and x = 6
O
B. x=-6 only
O
c. x= 6 only
O
D. x= -4 and x = 9
Answer:
c. x = 6 only
Step-by-step explanation:
In order to calculate the zeros of f(x), we need to set it equal to zero and find the corresponding values of x.
[tex]x^{2}-12x+36=0[/tex]
Using the midterm breaking, we can split -12x into two such terms whose sum will be -12x and product will be 36x². These two terms are -6x and -6x
So, the above expression can be written as:
[tex]x^2-6x-6x+36=0\\\\ x(x-6)-6(x-6)=0\\\\ (x-6)(x-6)=0\\\\ (x-6)^{2}=0\\\\ x-6=0\\\\ x=6[/tex]
This means, the zero of f(x) occurs at x = 6 only.
Find the equation of the axis of symmetry and the coordinates of the vertex of the graph of the function y=2x^2+4x-3
Answer:
Step-by-step explanation:
write this expression : f(x) = a(x-h)²+k
when the axis of symmetry is the line : x =h and the vertex A(h,k)
y=2x²+4x-3
y = 2(x²+2x - 3/2)
y=2((x²+2x+1) -1 -3/2)
y = 2((x+1)² - 5/2)
y = 2(x+1)² -5.....vertex form x= -1 the axis of symmetry and A(-1,-5)the vertex
Answer:
The equation of the axis of symmetry is x = -1.
The coordinates of the vertex are (-1, -5).
Step-by-step explanation:
y = 2x^2 + 4x - 3
y = 2(x^2 + 2x) - 3
y = 2[ (x + 1)^2 - 1] - 3
y = 2(x + 1)^2 - 5.
The equation of the axis of symmetry is x = -1.
The coordinates of the vertex are (-1, -5)
Solve the equation.
5/6 x-4 = -2
Answer:
2 2/5
Step-by-step explanation:
5/6 x-4 = -2
Add 4 to each side
5/6 x-4+4 = -2+4
5/6 x = 2
Multiply by 6/5 to isolate x
6/5 *5/6 x = 6/5 *2
x = 12/5
Changing this from an improper fraction
5 goes into 12 2 times with 2 left over
x = 2 2/5
He has 2yards of the string .He cut 5 /8 yards of the string for his project how much left
For this case we must subtract:
[tex]2- \frac {5} {8} =\\\frac {8 * 2-5} {8} =\\\frac {16-5} {8} =\\\frac {11} {8}[/tex]
Thus, you have [tex]\frac {11} {8}[/tex]of string, after you have cut[tex]\frac {5} {8}[/tex] of it.
If we convert to a mixed number we have to:
[tex]\frac {11} {8} = 1 \frac {3} {8}[/tex]
Answer:
He has [tex]1 \frac {3} {8}[/tex] of string.
Given the functions f(x) = 7x + 13 and g(x) = x + 2, which of the following functions represents f[g(x)] correctly? (2 points)
Answer:
The value of f[ g(x) ] = 7x + 27
Step-by-step explanation:
It is given that, f( x ) = 7x + 13 and g( x ) = x + 2
To find the value of f(g(x))
g(x) = x + 2 and 7x + 13 (given)
Let g(x) = x + 2
f [ g(x) ] = 7(x + 2) + 13 [ substitute the value of g(x) in f(x) ]
= 7x + 14 + 13
= 7x + 27
Therefore the value of f[ g(x) ] = 7x + 27
Subtract h + 3 from 6h + 1.
Enter your answer in the box.
Answer:
5h -2
Step-by-step explanation:
(6h + 1) - (h+3)
= 6h + 1 - h -3
= 5h -2
6h +1 - h+3
Subtract like terms:
6h - h = 5h
1-3 = -2
The answer becomes 5h-2
Six less than five times a number is at least thirty- four
a) solve the inequality
b) graph the solution on a number line
c) write the solution in interval notation
Step-by-step explanation:
Six less than five times a number is at least thirty- four
n - number
5n - 6 ≥ 34a)
5n - 6 ≥ 34 add 6 to both sides
5n ≥ 40 divide both sides by 5
n ≥ 8
b) in the attachment
c) n ∈ [8, ∞)
Which of the following functions has a graph that is a line?
f(x) = x
f(x) = x2
f(x) = |x|
Answer:
f(x) = x
Step-by-step explanation:
f(x) = x is of the form y = mx + b.
y = mx + b is the slope-intercept form of the equation of a line, so its graph is a line.
f(x) = x^2 has a variable to the second power, so its graph is a parabola, and not a line.
f(x) = |x| has an absolute value, so its graph is shaped like a V. It is not a line.
Gas costs $6 per gallon and diesel costs $8 per gallon. You have at most $85 to spend on fuel. You must purchase at least 12 gallons of gas for your car to run for the week. Let x be the amount of gas purchased and y be the amount of diesel purchased. Which of the following is a possible solution?
Answer:
(12,1.625)
Step-by-step explanation:
According to the given statement:
Cost of gas per gallon = $6
Cost of diesel per gallon = $8
Cost of x gallon gas = 6x
Cost of y gallon diesel = 8y
You have at most $85 to spend on fuel
Therefore the equation we get is:
6x+8y≤85
You must purchase at least 12 gallons of gas for your car
x≥12
Plot the graph and you get possible solutions:
(12,1.625)..
Find the inverse of 9^3/2
No Solution
Brainliest Please :)
Answer:
6
Step-by-step explanation:
9^3 ÷2 the inverse is 9÷3×2 =9÷3=3×2=6 the answer is 6
Graph the function in the coordinate plane. Use the Mark Feature tool to indicate the x- and y-intercepts of the function.y=2/3x+4
Answer:
My blue dot is the y-intercept.
My red dot is my x-intercept.
Please look at the graph.
Step-by-step explanation:
I can show you my graph and mark it where the x-intercepts and y-intercepts are.
Let's begin.
We have y=2/3 x+4.
Compare this to the slope-intercept form, y=mx+b where m is the slope and b is the y-intercept.
You should see that m=2/3 and b=4.
This means the slope is 2/3 and the y-intercept is 4.
Don't forget slope means rise/run.
So once we graph 4 (plot a point) on the y-axis, then we will use our slope to get to one more point. The slope here tells us to rise 2 and run 3.
Now sometimes our graph is not accurate when drawing by hand so there is a way without graphing that you can find the x- and y-intercepts.
The x-intercept is when the y-coordinate is 0.
The y-intercept is when the x-coordinate is 0.
So to find the x-intercept, I'm going to set y to 0 and solve for x. Like so,
0=2/3 x +4
Subtract 4 on both sides:
-4=2/3 x
Multiply both sides by the reciprocal of 2/3 which is 3/2:
3/2 (-4)=x
Simplify:
-12/2=x
Simplify:
-6=x
Symmetric Property:
x=-6
So the x-intercept is (-6,0).
I actually already have the y-intercept since my equation is in y=mx+b (slope-intercept form). But if it wasn't you could just set x to 0 and solve for y. Like so:
y=2/3 (0)+4
y=0+4
y=4
The y-intercept is (0,4).
Let's go to our graph now.
Answer:
After I had submitted my answer it gave me this answer
Step-by-step explanation:
Which points lie on the line that passes through point P and is parallel to the given line? Select three options.
(-4, 2)
(-1,3)
(-2, 2)
(4,2)
(-5, -1)
Answer:
So we have (-1,3), (-2,2) and (-5,-1).
Step-by-step explanation:
So we need to find the equation of the line that is parallel to the blue line going through point P.
The slope of the blue line (count the rise/run) is 1/1=1.
Parallel lines have the same slope.
So y=mx+b is the slope-intercept form where m is the slope and b is the y-intercept.
So since the blue line has slope 1 and parallel lines have the same slope then the line going through point P will have slope 1 too.
Point P is actually the y-intercept of the line going through P.
So the equation that is parallel to the blue line going through point P is y=1x+4. b was 4 because it was the y-intercept.
You can also just write y=1x+4 as y=x+4.
Testing the points:
(-4,2)?
Does y=x+4 for this point? 2=-4+4 which gives us 2=0 so no for this point.
(-1,3)?
Does y=x+4 for this point? 3=-1+4 which gives us 3=3 so yes for this point.
(-2,2)?
Does y=x+4 for this point? 2=-2+4 which gives us 2=2 so yes for this point.
(4,2)?
Does y=x+4 for this point? 2=4+4 which gives us 2=8 so no for this point.
(-5,-1)?
Does y=x+4 for this point? -1=-5+4 which gives -1=-1 so yes for this point.
So we have (-1,3), (-2,2) and (-5,-1).
Answer:BCE
Step-by-step explanation:I jus did it on EDGE
Marcel is nearing graduation. He just met with his counselor and she showed him this information.
Marcel’s College Costs & Payment Options per Year
Costs
Methods of Payment
Tuition & Fees
Grants & Scholarships
$12,500
$6,500
Room & Board
Work-Study
$8,200
$9,800
If he pays the balance with a student loan, how much will he need to borrow for his first year?
$4,400
$6,000
$10,900
$14,200
Answer:
The correct answer option is A. $4,400.
Step-by-step explanation:
Cost of the tuition and fees for 1 year = $12,500
Grants and scholarships for 1 year = $6500
Cost of room and board for 1 year = $8200
Cost for work study for 1 year = $9800
So basically, Marcel pays = (tuition and fees + cost of room and board)
= $12,500 + $8,200 = $20,700
He can pay through the (grants and scholarships + cost for work-study) = $6,500 + $9,800 = $16,300
Therefore, Marcel needs to borrow = $20,700 - $16,300 = $4,400
Answer:
The answer is A
Step-by-step explanation:
in the diagram below what is the approximate length of the minor arc DE
Answer:
C. 52 cm
Step-by-step explanation:
A full circle has a degree measure of 360 degrees.
Minor arc DE is intercepted by a central angle of 120 degrees.
120 degrees is 1/3 of 360 degrees, so the length of minor arc DE is 1/3 of the circumference of the circle.
length of minor arc DE = (120/360) * 2 * pi * r
length of minor arc DE = (1/3) * 2 * 3.14 * 25 cm
length of minor arc DE = 52.3333... cm
Answer:
the correct answer is 52.
Step-by-step explanation:
I just got it correct.
Select the statements that are true for the graph
The vertex is (-2, 4)-
The vertex is (2,4)
The graph has a minimum
The graph has a maximum
The graph has a y-intercept of -8
Answer:
The graph has a minimum.
Step-by-step explanation:
The top answer choice would also be a genuine statement if that negative symbol was removed from outside the 4. Second, whenever A is positive, the parabola opens up with a minimum value.
If you are ever in need of assistance, do not hesitate to let me know by subscribing to my channel [USERNAME: MATHEMATICS WIZARD], and as always, I am joyous to assist anyone at any time.