Which quadratic equation is equivalent to (x^2-1)^2-11(x^2-1)+24=0

Answer:

First case The coefficient of the squared term is 4

Second case The coefficient of the squared term is 1/16

Step-by-step explanation:

I will analyze two cases

First case (vertical parabola open upward)

we know that

The equation of a vertical parabola in vertex form is equal to

[tex]y=a(x-h)^{2}+k[/tex]

where

a is the coefficient of the squared term

(h,k) is the vertex

we have

(h,k)=(-5,-2)

substitute

[tex]y=a(x+5)^{2}-2[/tex]

Find the value of a

Remember that

when the x-value is -4, the y-value is 2.

substitute

For x=-4, y=2

[tex]2=a(-4+5)^{2}-2[/tex]

[tex]2=a(1)-2[/tex]

[tex]a=2+2=4[/tex]

the equation is equal to

[tex]y=4(x+5)^{2}-2[/tex]

therefore

The coefficient of the squared term is 4

Second case (horizontal parabola open to the right)

we know that

The equation of a horizontal parabola in vertex form is equal to

[tex]x=a(y-k)^{2}+h[/tex]

where

a is the coefficient of the squared term

(h,k) is the vertex

we have

(h,k)=(-5,-2)

substitute

[tex]x=a(y+2)^{2}-5[/tex]

Find the value of a

Remember that

when the x-value is -4, the y-value is 2.

substitute

For x=-4, y=2

[tex]-4=a(2+2)^{2}-5[/tex]

[tex]-4=a(4)^{2}-5[/tex]

[tex]-4+5=a(16)[/tex]

[tex]a=1/16[/tex]

the equation is equal to

[tex]x=(1/16)(y+2)^{2}-5[/tex]

therefore

The coefficient of the squared term is 1/16

to better understand the problem see the attached figure

Answer:

[tex]\large\boxed{x=-1\pm\sqrt5}[/tex]

Step-by-step explanation:

[tex]x^2+2x-4=0\qquad\text{add 4 to both sides}\\\\x^2+2x=4\\\\x^2+2(x)(1)=4\qquad\text{add}\ 1^2=1\ \text{to both sides}\\\\\underbrace{x^2+2(x)(1)+1^2}=4+1^2\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\(x+1)^2=5\to x+1=\pm\sqrt5\qquad\text{subtract 1 from both sides}\\\\x=-1\pm\sqrt5[/tex]

The two values of x for the given equation are ( -1  + √5 ) and ( - 1 - √5 ).

The polynomial having a degree of two or the maximum power of the variable in a polynomial will be 2 is defined as the quadratic equation and it will cut two intercepts on the graph at the x-axis.

The solution of the given equation is:-

=  x²  + 2x - 4

x²  +  2x  =  4

x²  +  2x ( 1 )  +  1²  =  4  +  1²

(  x  +  1  )²  =  5

x  +  1   =  ± √5

x  =  -1  ± √5

Therefore the two values of x for the given equation are ( -1  + √5 ) and ( - 1 - √5 ).

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Answer:

C. 3 times

Step-by-step explanation:

If cube a has an edge length of 2 and cube b has an edge length of 6,the volume of cube b than cube a is 3 times greater.

For this case we have that by definition, the volume of a cube is given by:

[tex]V = l ^ 3[/tex]

Where:

l: It's the side of the cube

Cube A:

[tex]l = 2\\V = 2 ^ 3 = 8 \ units ^ 3[/tex]

Cube B:

[tex]l = 6\\V = 6 ^ 3 = 216 \ units ^ 3[/tex]

We divide:

[tex]\frac {216} {8} = 27[/tex]

Thus, the volume of cube B is 27 times larger than that of cube A.

Answer:

Option A

Answer:

A = 44.2cm²

Step-by-step explanation:

The surface area of a square pyramid that has the sides of the base of 4 cm and a height of 10 cm is 44.2cm².

Formula: A=AB(4h2+AB)+AB

A=AB(4h2+AB)+AB=4·(4·102+4)+4≈44.1995cm²

Answer:

Inverse of a function

Step-by-step explanation:

If the x- and y-values in each pair of a set of ordered pairs are interchanged, the resulting set of ordered pairs is known as the inverse of a function.

For example, given the following function:

y = 2x

If x=0 → y= 0

If x=1 → y= 2

If x=2 → y= 4

Now, if we find the inverse of the function:  

y = 2x → x = 2y → y = x/2

Now:

If x=0 → y= 0

If x=2 → y= 1

If x=4 → y= 2

Comparing both cases, you will notice that the ordered pairs are effectively interchanged.

The resulting set of ordered pairs formed by interchanging the x- and y-values in each pair is known as the inverse of the original set.

If the x- and y-values in each pair of a set of ordered pairs are interchanged, the resulting set of ordered pairs is known as the inverse of the original set. In mathematics, an ordered pair is a pair of objects written in a specific order, typically as (x,y). If you switch the positions of the elements to form (y,x), you create an ordered pair that represents the inverse relationship. This is particularly relevant in the context of functions and relations on a Cartesian coordinate system. The concept of ordered pairs is fundamental to understanding mappings and the domain and range of relations.

For example, if you have an ordered pair representing a function, such as (3,4), its inverse would be (4,3). This reflects a new relationship where the original output is now the input and vice versa. The set of all such inverted pairs from a function forms the inverse function, provided that each y value is associated with only one x value (the function is one-to-one).

A relation where the ordering of elements matters contrasts with a set where the order does not affect the identity of the set. Hence, achieving an inverse relationship by switching the ordered pairs is a useful tool in mathematical problem-solving and analysis.

Answer:

The correct option is D) (5x − 2)(2x − 3).

Step-by-step explanation:

Consider the provided expression.

[tex]10x^2-19x+6[/tex]

Where x is time in minutes.

We need to find the appropriate form of the expression that would reveal the time in minutes when the trough is empty.

When the trough is empty the whole expression becomes equal to 0.

Substitute the whole expression equal to 0 and solve for x that will gives us the required expression.

[tex]10x^2-19x+6=0[/tex]

[tex]10x^2-15x-4x+6=0[/tex]

[tex]5x(2x-3)-2(2x-3)=0[/tex]

[tex](5x-2)(2x-3)=0[/tex]

Now consider the provided option.

By comparison the required expression is D) (5x − 2)(2x − 3).

Hence, the correct option is D) (5x − 2)(2x − 3).

Answer:

The correct answer is D

Step-by-step explanation:

Answer:

This is a right triangle

Step-by-step explanation:

Pythagoras theorem is used to determine if a triangle is right, acute or obtuse

If the sum of squares of two shorter lengths is greater than the square of third side then the triangle is an acute triangle.

If the sum of squares of two shorter lengths is less than the square of third side then the triangle is an obtuse triangle.

If the sum of squares of two shorter lengths is equal the square of third side then the triangle is a right triangle.

so,

[tex](17)^2 = (15)^2+(8)^2\\289 = 225+64\\289=289[/tex]

Therefore, the given triangle is a right triangle ..

Answer:

See below in bold,.

Step-by-step explanation:

There are 30-17 = 13 boys in the class.

a.  Prob(First is a boy ) = 13/30 and Prob( second is a boy = 12/29).

As these 2 events are independent:

Prob( 2 boys being picked) = 13/30 * 12/29 =  26/145 or 0.179 to the nearest thousandth.

b. By a similar method to a:

Prob ( 2 girls being picked) =  17/30 * 16/29 = 136/435 = 0.313 to the nearest thousandth.

c.  Prob (First student is a boy and second is a girl) =  13/30 * 17/29 = 221/870.

Prob ( first student is a girl and second is a boy)  = 17/30 * 13/29 = 221/870

These 2 events are not independent so they are added:

Prob( one of the students is a boy)  = 2 (221/870 =  221/435 = 0.508 to the nearest thousandth.

Answer:

Answer is all real numbers.

<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>

---------(-3)---------------------(6)-------------

Step-by-step explanation:

6b<36

Divide both sides by 6:

b<6

or

2b+12>6

Subtract 12 on both sides:

2b>-6

Divide both sides by 2:

b>-3

So we want to graph b<6 or b>-3:

               o~~~~~~~~~~~~~~~~~~~~~~~~~~ b>-3

~~~~~~~~~~~~~~~~~~~~~~~~o                        b<6

_______(-3)____________(6)___________

So again "or" is a key word! Or means wherever you see shading for either inequality then that is a solution to the compound inequality.  You see shading everywhere so the answer is all real numbers.

<~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>

---------(-3)---------------------(6)-------------

Answer:

All real numbers [tex](-\infty, \infty)[/tex]

Step-by-step explanation:

First we solve the following inequality

[tex]6b < 36[/tex]

Divide by 6 both sides of the inequality

[tex]b<\frac{36}{6}\\\\b<6[/tex]

The set of solutions is:

[tex](-\infty, 6)[/tex]

Now we solve the following inequality

[tex]2b + 12 > 6[/tex]

Subtract 12 on both sides of the inequality

[tex]2b + 12-12 > 6-12[/tex]

[tex]2b> -6[/tex]

Divide by 2 on both sides of the inequality

[tex]\frac{2}{2}b> -\frac{6}{2}[/tex]

[tex]b> -3[/tex]

The set of solutions is:

[tex](-3, \infty)[/tex]

Finally, the set of solutions for composite inequality is:

[tex](-\infty, 6)[/tex]  ∪ [tex](-3, \infty)[/tex]

This is: All real numbers [tex](-\infty, \infty)[/tex]

Answer:

56.4 miles

Step-by-step explanation:

4228/75 = 56.37333333

The coefficient of x needs to be 2.

The current coefficient is 2/5.

To eliminate the 5 in the denominator, we can multiply the equation by 5.

We get

[tex]2x + 30y = - 50[/tex]

Now, to solve the equations, all we need to do is add them.

Hope this helps!

[tex]x^2-x[/tex] is nothing interesting you can just factor out x and result with [tex]x(x-1)[/tex].

However the difference of squares is defined as,

[tex]a^2-b^2=(a+b)(a-b)[/tex]

Hope this helps.

r3t40

The value of the product (3 - 21) * (3 + 21) is -432, obtained by applying the distributive property.

To find the value of the product (3 - 21) * (3 + 21), we can use the distributive property or the difference of squares identity. Here's how it works:

(3 - 21) * (3 + 21) = (3 - 21) * [(3) + (21)]  (Apply the distributive property)

Now, let's calculate each part:

1. (3 - 21) = -18

2. (3 + 21) = 24

Now, we multiply these results together:

-18 * 24 = -432

So, the value of the product (3 - 21) * (3 + 21) is -432.

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Final answer:

The product of (3-21) and (3 + 21) is -324.

Explanation:

The value of the product (3-21) (3 + 21) is -324.

To find the product, first calculate the values within the parentheses:

3 - 21 = -18

3 + 21 = 24

Multiply these two values: -18 * 24 = -324.

Answer:

Step-by-step explanation:

It's a right triangle.

Use the Pythagorean theorem:

[tex]leg^2+leg^2=hypoyenuse^2[/tex]

We have:

[tex]leg=13.5\ cm,\ leg=x\ cm,\ hypotenuse=(x+8.45)\ cm[/tex]

Substitute:

[tex]13.5^2+x^2=(x+8.45)^2\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\182.25+x^2=x^2+2(x)(8.45)+8.45^2\qquad\text{subtract}\ x^2\ \text{from both sides}\\\\182.25=16.9x+71.4025\qquad\text{subtract 71.4025 from both sides}\\\\110.8475=16.9x\qquad\text{divide both sides by 16.9}\\\\\dfrac{110.8475}{16.9}=x\to x\approx6.6\ cm[/tex]

Answer:

1st and 2nd quadrants

Step-by-step explanation:

You have sine is positive since it is 4/5.

Sine is the y-coordinate.

On the coordinate plane y is positive in the 1st and 2nd quadrants.

The values of θ that could terminate when sin θ = 4/5 are in the first and second quadrants. In the first quadrant, both the sine and cosine values are positive, while in the second quadrant, only the sine value is positive.

To find the quadrants where θ could terminate when sin θ = 4/5, we need to examine the values of sin θ in each quadrant. The sine function is positive in the first and second quadrants so that θ could terminate in either of these quadrants. In the first quadrant, the sine and cosine values are positive, while in the second quadrant, only the sine value is positive.

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Answer:

[tex]\frac{x-1}{x+3}[/tex]

Step-by-step explanation:

Let's factor the numerator and denominator first.

x^2+5x-6 is a quadratic in the form of x^2+bx+c.

If you have a quadratic in the form of x^2+bx+c, all you have to do to factor is think of two numbers that multiply to be c and add to be b.

In this case multiplies to be -6 and adds to be 5.

Those numbers are 6 and -1 since -1(6)=-6 and -1+6=5.

So the factored form of x^2+5x-6 is (x-1)(x+6).

x^2+9x+18 is a quadratic in the form of x^2+bx+c as well.

So we need to find two numbers that multiply to be 18 and add to be 9.

These numbers are 6 and 3 since 6(3)=18 and 6+3=9.

So the factored form of x^2+9x+18 is (x+3)(x+6).

So we have that:

[tex]\frac{x^2+5x+-6}{x^2+9x+18}=\frac{(x-1)(x+6)}{(x+3)(x+6)}[/tex]

We can simplify this as long as x is not -6 as

[tex]\frac{x-1}{x+3}[/tex]

I obtained the last line there by canceling out the common factor on top and bottom.

Answer:

We can simplify this as long as x is not -6 as

\frac{x-1}{x+3}

Step-by-step explanation:

Answer:

x = [tex]\frac{7 - r}{a}[/tex]

Step-by-step explanation:

ax+r=7

ax = 7 - r

x = [tex]\frac{7 - r}{a}[/tex]

Answer:

Option B, C and D are correct choices.

Step-by-step explanation:

We have been given an equation [tex]3.5+1.2(6.3-7x)=9.38[/tex]. We are asked to choose the steps that are involved in solving the equation.

Let us solve the equation.

Using distributive property [tex]a(b+c)=ab+ac[/tex], we will distribute 1.2 to 6.3 and [tex]-7x[/tex].

[tex]3.5+1.2*6.3-1.2*7x=9.38[/tex]

[tex]3.5+7.56-8.4x=9.38[/tex]

Therefore, option B is the correct choice.

Now, we will combine like terms.

[tex]11.06-8.4x=9.38[/tex]

Therefore, option C is the correct choice.

Now, we will subtract 11.06 from both sides of our equation.

[tex]11.06-11.06-8.4x=9.38-11.06[/tex]

[tex]-8.4x=-1.68[/tex]

Therefore, option D is the correct choice.

Now, to solve for x, we need to divide both sides of our equation by [tex]-8.4[/tex]

[tex]\frac{-8.4x}{-8.4}=\frac{-1.68}{-8.4}[/tex]

[tex]x=0.2[/tex]

Answer:

Distribute 1.2 to 6.3 and –7x;

Combine 3.5 and 7.56.

Subtract 11.06 from both sides.

Step-by-step explanation:

To answer this expression. Let's follow P.E.M.D.A. order, the acronym for  PArenthesis, Exponents, Multiplication, Division and Addends.  So distributing the factor 1.2 to the parenthesis content:

[tex]3.5+1.2(6.3-7x)=9.38 \\3.5+7.56-8.4x=9.38[/tex]

Then adding the 3.5 to 7.56 to simplify it:

[tex]3.5+7.56-8.4x=9.38\\\11.06-8.4x=9.38[/tex]

The next step in order to isolate is to subtract 11.06 from both sides

[tex]11.06-8.4x-11.06=9.38-11.06[/tex]

Then it goes on

[tex]-8.4x=-1.68\:\:*(-1)\\8.4x=1.68\Rightarrow x=\frac{1.68}{8.4}\Rightarrow x=\frac{1}{5}\\S=\{ {\frac{1}{5}\}[/tex]

Answer:

6

Step-by-step explanation:

I took the same test

Answer:

2.95

Step-by-step explanation:

The radius of a circle is half of the diameter.

r = (1/2)d

r = (1/2)(5.9)

r = 2.95

Answer:

radius = 2.95

Step-by-step explanation:

The radius is the distance from the centre O to the circumference and is one half the diameter,

radius = 0.5 × 5.9 = 2.95

Answer:

The slope of XZ is 3/4 , the slope of XY is -4/3 , and XZ = XY = 5 ⇒ 3rd answer

Step-by-step explanation:

* Lets look to the attached figure to solve the problem

- To prove that the Δ XYZ is an isosceles right triangle, you must

  find two sides the product of their slopes is -1 and they are equal

  in lengths

- From the figure the vertices of the triangle are;

  X = (1 , 3) , Y = (4 , -1) , Z = (5 , 6)

- The slope of the line whose endpoints are (x1 , y1) and (x2 , y2)

  is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

∵ The slope of [tex]XY=\frac{-1-3}{4-1}=\frac{-4}{3}[/tex]

∵ The slope of [tex]XZ=\frac{6-3}{5-1}=\frac{3}{4}[/tex]

∴ The slope of XY = -4/3 , the slope of XZ = 3/4

∵ -4/3 × 3/4 = -1

∴ XY ⊥ XZ

∴ ∠ X is a right angle

∴ Δ XYZ is a right triangle

- The distance between the two points (x1 , y1) and (x2 , y2) is

  [tex]d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[/tex]

∵ [tex]XY=\sqrt{(4-1)^{2}+(-1-3)^{2}}=\sqrt{9+16}=\sqrt{25}=5[/tex]

∵ [tex]XZ=\sqrt{(5-1)^{2}+(6-3)^{2}}=\sqrt{16+9}=\sqrt{25}=5[/tex]

∴ XY = XZ = 5

∴ Δ XYZ is an isosceles right triangle

* The statement which prove that is:

 The slope of XZ is 3/4 , the slope of XY is -4/3 , and XZ = XY = 5  

Answer:

its C

Step-by-step explanation:

yw :)

Answer:

(1/7)(-9n + 7)

Step-by-step explanation:

1/7-3(3/7n-2/7)  can be factored:  factor out 1/7.

We get:  (1/7)[1 - 3(3n - 2)], or

              (1/7)[1 - 9n + 6], or

              (1/7)(-9n + 7)

The boy bought 9 pencils and 9 pens, with a total cost of 90 soles. This calculation is based on the cost of 3 soles per pencil and 7 soles per pen, resulting in a total expenditure that aligns with the given information.

1: Define the costs.

We know the cost of each pencil and pen:

Pencils: 3 soles each

Pens: 7 soles each

2: Let x be the number of items.

Since the boy buys the same number of pencils and pens, let x represent the number of each item he buys.

3: Set up the total cost equation.

The total cost he spends is 90 soles. We can express this as an equation:

Total cost = Pencils cost + Pens cost

90 soles = (x pencils * 3 soles/pencil) + (x pens * 7 soles/pen)

4: Simplify and solve for x.

Combine like terms:

90 soles = 3x soles + 7x soles

90 soles = 10x soles

x = 90 soles / 10 soles/item

x = 9 items

5: Find the number of pencils and pens.

Since x represents both the number of pencils and pens, he bought 9 pencils and 9 pens.

Therefore, the boy bought 9 pencils and 9 pens.

Complete question:

A boy buys the same number of pencils as pencils for 90 soles. Each pencil costs 3 soles and each pencil 7 soles. How many pencils and pens has he bought?

Answer:

A plane figure with 4 sides is called a quadrilateral.

A quadrilateral is a plane figure bounded by four straight lines in mathematics.

Plane figure bounded by four straight lines: In mathematics, a quadrilateral is a plane figure bounded by four straight lines. Examples of quadrilaterals include squares, rectangles, parallelograms, and trapezoids.

Answer:

Step-by-step explanation:

[tex]\text{Because}\\\\125x^3=5^3x^3=(5x)^3\qquad\bold{it's\ a\ cube}\\\\169=13^2\qquad\bold{it's \ a \ square}\\\\125x^3+169=(5x)^3+13^2[/tex]

[tex]13-\text{it's a prime number}[/tex]

Answer:

0, 17

3 1/2, 5

-3 1/2, -12

0, 17

Answer:

17

Step-by-step explanation:

First label the 2 points.

A = (3, 1/2, 5), B = (3, 1/2, -12)

Then calculate the vector from A to B:

[tex]\vect{AB} = (0, 0, -17)[/tex]

And then calculate it's length by the formula:

[tex]||\vect{a}|| = \sqrt{x^2 + y^2 + z^2}[/tex], where x, y, z are the coordinates related to the vector.

[tex]||\vect{AB}|| = \sqrt{0^2 + 0^2 + (-17)^2} = \sqrt{(-17)^2} = |17| = 17[/tex]

The correct solution is,

⇒ 2 / (1 - tan²x)

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

Given that;

The expression is,

⇒ tan (2x) / tan x

Now, We know that;

tan 2x = (2 tanx / 1 - tan²x)

Hence, We can simplify as;

⇒ tan (2x) / tan x

⇒ (2tan (x) /1 - tan²x) / tan x

⇒ 2 / (1 - tan²x)

Thus, The correct solution is,

⇒ 2 / (1 - tan²x)

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The expression tan(2x)/tan(x) simplifies using the double-angle identity for tangent to 2/(1 - tan^2(x)), which corresponds to Option A.

To simplify the trigonometric expression tan(2x)/tan(x) using double-angle identities, we can use the double-angle identity for tangent:

tan(2x) = 2tan(x)/(1 - tan2(x)).

Now, if we divide tan(2x) by tan(x), we get:

tan(2x)/tan(x) = (2tan(x)/(1 - tan2(x)/tan(x).

With tan(x) in the numerator and denominator, it cancels out, leaving:

2/(1 - tan2(x)).

Therefore, the correct answer is Option A: 2/(1 - tan2(x)).

Answer:

Ok

Step-by-step explanation: what is this???????

let's recall Cavalieri's Principle, solids with equal altitudes and cross-sectional areas at each height have the same volume, so even though this cylinder is slanted with a height = x and a radius = 8, the cross-sectional areas from the bottom to top are the same thickness and thus the same area, so its volume will be the same as a cylinder with the same height and radius that is not slanted.

[tex]\bf \textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\ \cline{1-1} r=8\\ h=x\\ V=768\pi \end{cases}\implies 768\pi =\pi (8)^2(x)\implies 768\pi =64\pi x \\\\\\ \cfrac{768\pi }{64\pi }=x\implies 12=x[/tex]

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.