The first six steps of the derivation of the Quadratic Formula are listed below, find the 7th,8th, and 9th steps
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Final answer:
The derivation of the quadratic formula involves several steps, including identifying the unknown, identifying the knowns, choosing an equation, finding additional equations if necessary, solving simultaneous equations, eliminating terms, substituting known quantities, and checking the answer for reasonableness.
Explanation:
Derivation of the Quadratic Formula
Identify the unknown.Identify the knowns.Choose an equation, plug in the knowns, and solve for the unknown.Find an equation or set of equations that can help you solve the problem. If there is more than one unknown, additional equations may be necessary.Solve the simultaneous equations for the unknowns.Eliminate terms wherever possible to simplify the algebra.Substitute the known quantities, along with their units, into the appropriate equation and obtain numerical solutions complete with units.Check the answer to see if it is reasonable: Does it make sense?The seventh step is to take the square root of both sides. The eighth step is to isolate x by subtracting [tex]\(\frac{B}{2a}\)[/tex]. The ninth step simplifies the expression.
The seventh step is:
7. Take the square root of both sides to isolate x:
[tex]\[ x + \frac{B}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} \][/tex]
The eighth step is:
8. Solve for x by subtracting [tex]\(\frac{B}{2a}\)[/tex] from both sides:
[tex]\[ x = -\frac{B}{2a} \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} \][/tex]
The ninth step is:
9. Simplify and solve for x:
[tex]\[ x = \frac{-B \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
This is the quadratic formula, which gives the solutions for the equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
Related Questions
A scientist is working with 2 meters of gold wire. How long is the wire in millimeters? Be sure to include the correct unit in your answer. ( mm, cm, dm, m, km )
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2 meters(m) = 2000 millimeters (mm)
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Have a great day!
The correct answer is 2000 mm.
To convert the length of the gold wire from meters to millimeters, one must know the conversion factor between these two units. There are 1000 millimeters in 1 meter. Therefore, to convert 2 meters to millimeters, one multiplies the length in meters by 1000.
Calculation:
2 meters * 1000 millimeters/meter = 2000 millimeters
Thus, the length of the gold wire in millimeters is 2000 mm.
A ________ is a monomial or the sum or difference of two or more monomials. Base, polynomial, linear term,quadratic equation
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"A polynomial is a monomial or the sum or difference of two or more monomials".
By definition, the terms of a Polynomial Function have the form ax^n, where "n" is a natural number.
To classify a polynomial, you must count the number of its terms. For example:
5x²-2x (This is a binomial, because it is formed by two terms).
x³-3x+1 (This is a trinomial, because it has three terms):
To identify the degree of a polynomial, you must see the term which has the highest exponent. For example, the degree of following polynomial is 2.
4x²-x+3
The degree of this polynomial is 3:
x³+2x²-x+10
y=-x+7 , y=x-1 and show your work Help !
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[tex]7=2x-1 \\ 8=2x \\ 4=x[/tex]
Then, substitute 4 into one of the original equations to find y.
[tex]y=(4)-1=3[/tex].
The solution is [tex](4,3)[/tex]. I have attached a graphical representation of the solution as well.
Marci is going to use tile to cover her terrace. How much tile does she need?
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The answer is 4*8+(16p/2)
32+8p
A=(32+8p)m^2
A=57.13 m^2 (to nearest hundredth of a m^2)
Hope this helps you>:)
~ TRUE BOSS
16 is what fractional part of 20
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The number 16 is 4/5 fractional part of 20.
What is mean by Fraction?
A fraction is a part of whole number, and a way to split up a number into equal parts. Or, A number which is expressed as a quotient is called fraction. It can be written as the form of p : q, which is equivalent to p / q.
Let the fraction part of 16 = x
So, We can formulate;
⇒ 16 = x × 20
⇒ x = 16 / 20
⇒ x = 4/5
Thus, The number 16 is 4/5 fractional part of 20.
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shannon uses 30 inches of ribbon to make one bow.how many feet are needed to make 10 bows?
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hope this helps
PLEASE HELP ME WITH THE REST OF THE QUESTIONS
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The puzzles measures 12 inches by 12 inches.what is the area of the puzzle.
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you are asked to determine if the two events "a randomly selected student has asthma" and "a randomly selected student has a household member who smokes" are independent. what probabilities could you calculate to answer this question?
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event A = student has asthma
event B = at least one household member smokes
To find the probability of event A happening, we count the total number of people who have asthma out of the total in the survey. So we simply look at the "student has asthma" row and look at the "total" column. This value is 182. There are 182 people who have asthma. This is out of 937 (bottom right corner).
Divide the values: 182/937 = 0.194237
This value is approximate
So P(A) = 0.194237 approximately
The P means probability
P(A) = probability of event A happening
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Now compute P(B)
P(B) = 395/937
P(B) = 0.421558
Notice I'm dividing the total value in the "at least one member smokes" column over the grand total of 937
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Next we compute P(A and B) which is the probability both events occur at the same time.
P(A and B) = 113/937
P(A and B) = 0.120598
The 113 is the amount of people who have asthma AND have at least one household member that smokes (row1, column2)
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If events A and B were independent, then P(A)*P(B) should be equal to P(A and B)
Let's see if that is the case
P(A and B) = P(A)*P(B)
0.120598 = P(A)*P(B)
0.120598 = 0.194237*0.421558
0.120598 = 0.08188216
we get two different results so P(A and B) = P(A)*P(B) is not true
Therefore, A and B are not independent events. They are said to be dependent events.
9. 10. 11. And need work or won’t get credit
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Answer:
9) 22 - 5(6v - 1) = -63
22 - 30v + 5 = -63
27 - 30v = -63
-30v = -90
v = 3
10) 18x - (8x - 7) = 67
18x - 8x + 7 = 67
10x = 60
x = 7
11) 8(-2x - 4) + 12 = -52
8(-2x - 4) = -64
-2x - 4 = -8
-2x = -4
x = 2
A machinist creates a washer by drilling a hole through the center of a circular piece of metal. If the piece of metal has a radius of x + 7 and the hole has a radius of x + 6, what is the area of the washer?
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[tex]A_c = \pi r^2[/tex]
Now, we need to find the radius of the entire metal circle which the hole was drilled on:
[tex]R_w = (x+7) - (x+6) = 1[/tex]
Plug it in to find the area of the metal plate:
[tex]A_p = \pi (1)^2 = \pi [/tex]
Now, you subtract the area of the hole from the area of the plate. The area of the hole will be:
[tex]A_h = \pi (x+6)^2[/tex]
[tex]A_h = \pi (x^2+12x+36) = [/tex]
Subtract the area of the hole from the metal plate to get:
[tex]A_w = \pi - (\pi x^2+12 \pi x+36 \pi) = \pi x^2+12 \pi x -35 \pi[/tex]
That's your answer, although it looks wrong because not much information was provided.
Help me with this one question and think closely it is not 8 or 800 question 36
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≈ 22.75 gal
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The conversion from liters to gallons is accurate to the digits shown, but is not exact.
The table shows the change in the low temperature in Albany compared to the day before.
Wednesday : +3.5
Thursday: -4.1
Friday : 0
Saturday: -2.2
On which day did the low temperature change the most?
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Answer: On Thursday the low temperature change the most.
Explanation :
From the given table we can conclude that change is the lowest in Thursday .
And there is no matter if it is positive or negative
Because change defines final - initial only i.e. difference between final and initial
So, we consider absolute value only.
According to which Thursday has the atmost change i.e 4.1 which is the lowest amongst the other .
So ,on Thursday the low temperature change the most.
The day with the most temperature change in Albany compared to the day before is Thursday.
The day with the most temperature change was Wednesday.
To determine the most significant change, we calculate the absolute difference for each day's temperature change:
Wednesday: |+3.5| = 3.5Thursday: |-4.1| = 4.1Friday: |0| = 0Saturday: |-2.2| = 2.2Therefore, Thursday had the most substantial temperature change.
If Jimmy increases a number by 10%, and then decreases this new value by 10%, what percent of the original number will the final result be?
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Then he reduces it by 10% of the new number, which s 11%.
110% - 11% = 99%
The answer is 99%.
Hope this helps!
An investment of $900 is invested at a rate of 4%. The function that models this situation if f(t)=900(1+0.4^t) where does x represent time in years what is the rate of time in years. What is the rate of change of interval (3,10)
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To find the rate of change of the interval (3,10) for the given function, you need to calculate the difference in the function values at those two time points and divide it by the difference in time. The rate of change of the interval (3,10) is: ((900(1 + 0.4^10)) - (900(1 + 0.4^3))) / (10 - 3)
Explanation:The given function f(t) = 900(1 + 0.4^t) models an investment of $900 at a rate of 4% over time, where t represents time in years. To find the rate of change of the interval (3,10), you need to find the difference in the function values at those two time points and divide it by the difference in time.
So, the rate of change of the interval (3,10) is:
((900(1 + 0.4^10)) - (900(1 + 0.4^3))) / (10 - 3)
How many solutions to the equation
5(x-3)-3x=8x-15-6x
A) no solution
B) excalty no solution
C) excalty two solution
D) infinitely many solutions
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Since both sides equal, they are infinitely many solutions
D) infinitely many solutions
Matthew has saved a total of 10400 of which 2600 is invested in the stock market. What percent of his total saving is invested in the stock market
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Sorry if this is incorrect
What is the same as 98.52 divided by 10
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Dividing 98.52 by 10 involves moving the decimal one place to the left, resulting in 9.852. This process is the same for all decimals when dividing by powers of 10, such as 10, 100, or 1000.
When we divide a number by powers of 10, we're effectively moving the decimal point to the left by as many places as there are zeros in the power of ten. For instance, dividing 1.9436 by 10 moves the decimal one place to the left, becoming 0.19436. Similarly, dividing 98.52 by 10 moves the decimal one place to the left as well, giving us the simplified answer of 9.852.
If we were to multiply a number by powers of 10, we would do the opposite, moving the decimal to the right. For example, to multiply 45.86 by 100, you would move the decimal two places to the right to get 4586.
In conclusion, 98.52 divided by 10 is 9.852. Even if the answer is rounded, the process remains the same; you simply adjust the decimal point accordingly based on the power of 10 you are dividing by.
You are a farmer and want to spend under $35,000 on farm equipment. You need a hay bailer that costs $6,250 and several plowing disks cost $2,500 each. Write an inequality that models how many plowing disks could be purchased within your budget. What is the maximum number of plowing disks you can buy?
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Maximum plowing disks: 10 which would cost you $31, 250
Answer:
A) 2,500x + 6,250 < 35,000; 11 plowing disks
Step-by-step explanation:
PLZ HELP I BEG U IT MAKE NO SENSE
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Principle = $15,500 Interest Rate = 8.75% Time = ? Years Simple Interest = $5425.00
Solve for the Time in Years
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I = P x i x t
I is the interest
P is the total principal
i is the rate of interest per year
t is the total time in years
In this example, when we plug our known values in, our formula looks like this:
t = [tex] \frac{I}{i*p} [/tex]
t = [tex] \frac{5425}{0.0875 * 15500} [/tex]
t = 4
So time (in years) would be 4 years.
Helpppp!!! ASAP!!!!!!!!
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PLEASE ANSWER FAST!!!!
A rectangular prism has a length of 114 centimeters, a width of 4 centimeters, and a height of 314 centimeters. What is the volume of the prism? Enter your answer in the box as a simplified mixed number or a decimal.
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200 grams of fertilizer are used for each tree in an orchard containing 1200 trees. at $2.75 per kilogram of fertilizer, how much does it cost to fertilize the orchard ?
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To fertilize an orchard of 1200 trees, with 200 grams of fertilizer per tree and the cost of fertilizer at $2.75 per kilogram, the total cost would be $660.
To calculate the cost to fertilize an orchard of 1200 trees, assuming 200 grams of fertilizer are used per tree, and the fertilizer costs $2.75 per kilogram, we need to follow these steps:
Convert the amount of fertilizer needed per tree into kilograms, as the cost is given per kilogram. Since there are 1000 grams in a kilogram, 200 grams is equal to 0.2 kilograms.
Multiply the amount of fertilizer needed per tree by the number of trees to find the total amount of fertilizer needed for the orchard. This is 0.2 kg/tree times 1200 trees = 240 kilograms.
Multiply the total kilograms of fertilizer by the cost per kilogram to find the total cost of fertilizing the orchard. This is 240 kg times $2.75/kg = $660.
Therefore, the total cost to fertilize the orchard is $660.
Landons front lawn measures 20 feet by 8 feet he counts to find that there are 120 blades of grass per square inch how many blade do grass are in the entire lawn use proportions
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1 foot = 12 inches
20feet = 20*12=240 inches
8feet = 8*12=96 inches
Now we can calculate total area:
A = height * width
A= 240 * 96
A=23040 inches²
We know the amount per one inch² and we need to find out total number of blades:
1 inch² 120 blades
23040 inches² x blades
Now we cross-multiply numbers:
1*x = 23040 * 120
x= 2764800
There are total of 2764800 blades in the entire lawn
I believe he area is incorrect. So can somebody help me to figure out the area and arc length. PLEASE show your work. So I can understand :)
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To find the arc length, you must use the formula: [tex] \frac{arc degree}{360} * 2 \pi r[/tex], then we plug in 80 as our arc degree and 1.3 as our radius, and we get: 1.81514242207
solve for k
j= k^2/4
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4j=k^2
k=sqrt(4j) or (4j)^1/2
Answer:
k = +- 2sqrt(J)
Step-by-step explanation:
This is the correct answer, I just took the test and this is right!
A cone has a radius of 11 inches and a height of 4.5 inches.
What is the volume of the cone?
Use 3.14 for pi. Round to the nearest tenth.
A. 49.5 in³
B. 121.4 in³
C. 569.9 in³
D. 1024.4 in³
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Answer:
Step-by-step explanation:
Volume = πr2h/3
radius of the cone is 11 inches
height of the cone is 4.5 inches
V= (3.14 *(121)*4.5)/3= 569.91 inches3
Let ∠1, ∠2, and ∠3 have the following relationships.
∠1 and ∠2 are acute vertical angles.
∠3 is an obtuse angle adjacent to both ∠1 and ∠2.
What is the sum of the measure of ∠1 and the measure of ∠3?
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"The value of [tex]\angle1 +\angle3[/tex] lies between [tex]90^{\circ} < \angle1+\angle3< 180^{\circ}[/tex].
Given,[tex]\angle 1,\angle2, \angle3[/tex] have the following relationships.
[tex]\angle 1[/tex] and[tex]\angle2[/tex] are acute vertical angles. [tex]\angle 3[/tex] is an obtuse angle adjacent to both [tex]\angle 1[/tex] and [tex]\angle 2[/tex].We have to find the sum of measure of [tex]\angle1[/tex] and the measure of [tex]\angle3[/tex].
Angle sum property:We know that the angle sum property of triangle states that the sum of all angles of a triangle is equal to [tex]180 ^{\circ}[/tex].
[tex]\angle 1+\angle 2+\angle3= 180^ {\circ}[/tex]
[tex]\angle1+\angle3= 180-\angle2[/tex]
So, the value [tex]\angle1 and \angle3[/tex] is always greater than [tex]90{^\circ}[/tex] because [tex]\angle 3[/tex] is the obtuse angle and it is being added to [tex]\angle1[/tex],
so it is always greater than right angle and less than [tex]180^{\circ}[/tex].
Hence the value of [tex]\angle1 +\angle3[/tex] lies between [tex]90^{\circ} < \angle1+\angle3< 180^{\circ}[/tex].
For more details on angle sum property follow the link:
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using similarity solve for x. show all work
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First find h.
h²= 9 x 16
h²=144
Take square root.
h=12
Now to find x, we use Pythagorean theorem.
c²=a²+b²
Put values
x²=9²+12²
x²=81+144
x²=225
Take square root.
x=15
Answer: x=15