How to factor an equation - ACT Math

Card 0 of 12

Question

If (x2 + 2) / 2 = (x2 - 6x - 1) / 5, then what is the value of x?

Answer

(x2 + 2) / 2 = (x2 - 6x - 1) / 5. We first cross-multiply to get rid of the denominators on both sides.

5(x2 + 2) = 2(x2 - 6x - 1)

5x2 + 10 = 2x2 - 12x - 2 (Subtract 2x2, and add 12x and 2 to both sides.)

3x2 + 12x + 12 = 0 (Factor out 3 from the left side of the equation.)

3(x2 + 4x + 4) = 0 (Factor the equation, knowing that 2 + 2 = 4 and 2*2 = 4.)

3(x + 2)(x + 2) = 0

x + 2 = 0

x = -2

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Question

Solve 8x2 – 2x – 15 = 0

Answer

The equation is in standard form, so a = 8, b = -2, and c = -15. We are looking for two factors that multiply to ac or -120 and add to b or -2. The two factors are -12 and 10.

So you get (2x -3)(4x +5) = 0. Set each factor equal to zero and solve.

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Question

If , and , which of the following is a possible value of ?

Answer

The given expression is a quadratic equation; therefore, we can factor the equation

Use the format of the standard quadratic equation:

Since , we know that the quadratic's roots will resemble the following:

$(m + \ )(m -\ ) = 0$

We also know that one of those signs has to be negative, since our two last terms multiply to equal the variable , and is negative in our quadratic. Now, we need to find two numbers that when multiplied together equal -24, and equal 10 when they are added together. Let's start by finding the factors of 24. The factors of 24 are 24 and 1, 12 and 2, 8 and 3, 6 and 4. Since one of those factors will be negative in our factored equation, we need to find the two factors whose difference is 10.

This means that the numbers in the factored equation are 12 and -2; thus, we may write the following:

.

By the zero multiplication rule, either portion of that equation can equal 0 for the result to be 0; thus, we have the following two expressions:

Subtract 12 from both sides of the equation:

Calculate the value of the variable in the second equation.

Add 2 to both sides of the equation.

Since we want a negative answer for our variable, the correct answer is:

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Question

What is the value of where:

Answer

The question asks us to find the value of , because it is in a closed equation, we can simply put all of the whole numbers on one side of the equation, and all of the containing numbers on the other side.

We utilize opposite operations to both sides by adding to each side of the equation and get

Next, we subtract from both sides, yielding

Then we divide both sides by to get rid of that on

$x=\frac{1}{5}$

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Question

For what value of b is the equation b2 + 6b + 9 = 0 true?

Answer

Factoring leads to (b+3)(b+3)=0. Therefore, solving for b leads to -3.

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Question

What is the solution to:

$\frac{x^2-6x+8}{x-2}=0$

Answer

First you want to factor the numerator from x2 – 6x + 8 to (x – 4)(x – 2)

Input the denominator (x – 4)(x – 2)/(x – 2) = (x – 4) = 0, so x = 4.

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Question

Which of the following is a factor of the polynomial x_2 – 6_x + 5?

Answer

Factor the polynomial by choosing values that when FOIL'ed will add to equal the middle coefficient, 3, and multiply to equal the constant, 1.

x_2 – 6_x + 5 = (x – 1)(x – 5)

Because only (x – 5) is one of the choices listed, we choose it.

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Question

Factor the following equation:

${x^3-2x^2+x}$

Answer

First we factor out an x then we can factor the

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Question

7 times a number is 30 less than that same number squared. What is one possible value of the number?

Answer

$\small 7x+30=x^{2}$

$\small x^{2}-7x-30=0$

$\small (x-10)(x+3)=0$

Either:

$\small x-10=0$

$\small x=10$

or:

$\small x+3=0$

$\small x=-3$

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Question

Which of the following is equivalent to $2x^{2}\left ( xy^{2}+5x^{2}y^{2} \right )$ ?

Answer

The answer is $2x^{3}y^{2}+10x^{4}y^{2}$ .

To determine the answer, $2x^{2}$ must be distrbuted,

$(2x^{2}xy^{2}) +(2x^{2}5x^{2}y^{2})$ . After multiplying the terms, the expression simplifies to $2x^{3}y^{2}+10x^{4}y^{2}$ .

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Question

Which of the following equations is NOT equivalent to the following equation?

$4y^{2}=169x^{2}-81$

Answer

The equation presented in the problem is:

$4y^{2}=169x^{2}-81$

We know that:

$169x^{2}-81=(13x)^{2}-9^{2}=(13x+9)(13x-9)$

Therefore we can see that the answer choice $(2y)^{2}=(13x+9)(13x-9)$ is equivalent to $4y^{2}=169x^{2}-81$ .

$\frac{(2y)^{2}}{13x+9}=27x-9-14x$ is equivalent to $4y^{2}=169x^{2}-81$ . You can see this by first combining like terms on the right side of the equation:

$\frac{(2y)^{2}}{13x+9}=13x-9$

Multiplying everything by , we get back to:

$(2y)^{2}=(13x+9)(13x-9)$

We know from our previous work that this is equivalent to $4y^{2}=169x^{2}-81$ .

$6y^{2}=\frac{3\times (13x+9)\times (13x-9)}{2}$ is also equivalent $4y^{2}=169x^{2}-81$ since both sides were just multiplied by $\frac{3}{2}$ . Dividing both sides by $\frac{3}{2}$ , we also get back to:

$(2y)^{2}=(13x+9)(13x-9)$ .

We know from our previous work that this is equivalent to $4y^{2}=169x^{2}-81$ .

$y=\sqrt{\frac{338x^{2}-162}{8}}$ is also equivalent to $4y^{2}=169x^{2}-81$ since

$y^{2}=(\sqrt{\frac{338x^{2}-162}{8}})^{2}=\frac{338x^{2}-162}{8}$

$4y^{2}=\frac{338x^{2}-162}{2}=169x^{2}-81$

Only $y^{2}=(\frac{13x-9}{2})^{2}$ is NOT equivalent to $4y^{2}=169x^{2}-81$

because

$y^{2}=(\frac{13x-9}{2})^{2}=\frac{169x^{2}-234x-81}{4}$

$4y^{2}=169x^{2}-81-234x eq 4y^{2}=169x^{2}-81$

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Question

A certain number squared, plus four times itself is equal to zero.

Which of the following could be that number?

Answer

The sentence should first be translated into an equation.

We will call the "certain number" .

This gives us $x^{2}+4x=0$ .

So solve this we can factor out an to get .

This makes the zeros of the equation evident. We know that when the entire expression will equal zero, making the equation true. We also know that when , the quantity will be , which also satisfies the equation.

Therefore, the two possible solutions are and .

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