ACT Math : How to factor an equation

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Factor An Equation

Solve 8x2 – 2x – 15 = 0

Possible Answers:

x = 3/2 or -5/4

x = -3/2 or 5/4

x = 3/2 or 5/4

x = -3/2 or -5/4

Correct answer:

x = 3/2 or -5/4

Explanation:

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.

 

 

Example Question #1 : How To Factor An Equation

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

Possible Answers:

2

4

-2

-3

3

Correct answer:

-2

Explanation:

(x+ 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

 

Example Question #1 : Factoring Equations

Which of the following is a factor of the polynomial x2 – 6x + 5?

Possible Answers:

x – 8

x + 2

x + 1

x – 6

x – 5

Correct answer:

x – 5

Explanation:

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.

x2 – 6+ 5 = (x – 1)(x – 5)

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

Example Question #1 : How To Factor An Equation

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

Possible Answers:

3\(\displaystyle 3\)

0\(\displaystyle 0\)

-10\(\displaystyle -10\)

1\(\displaystyle 1\)

-3\(\displaystyle -3\)

Correct answer:

-3\(\displaystyle -3\)

Explanation:

\small 7x+30=x^{2}\(\displaystyle \small 7x+30=x^{2}\)

\small x^{2}-7x-30=0\(\displaystyle \small x^{2}-7x-30=0\)

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

Either:

\small x-10=0\(\displaystyle \small x-10=0\)

\small x=10\(\displaystyle \small x=10\)

or:

\small x+3=0\(\displaystyle \small x+3=0\)

\small x=-3\(\displaystyle \small x=-3\)

Example Question #2 : How To Factor An Equation

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

Possible Answers:

\(\displaystyle 2x^{3}y^{2}+x^{4}y^{2}\)

\(\displaystyle 2x^{3}y^{2}+10x^{4}y^{2}\)

\(\displaystyle 2x^{3}y^{2}+10x^{4}y\)

\(\displaystyle 2xy^{2}+x^{4}y^{2}\)

\(\displaystyle x^{3}y^{2}+10x^{4}y^{2}\)

Correct answer:

\(\displaystyle 2x^{3}y^{2}+10x^{4}y^{2}\)

Explanation:

The answer is \(\displaystyle 2x^{3}y^{2}+10x^{4}y^{2}\).

To determine the answer, \(\displaystyle 2x^{2}\) must be distrbuted,

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

Example Question #2 : Factoring Equations

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

Possible Answers:

3

0

3

5

Correct answer:

3

Explanation:

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

Example Question #3 : How To Factor An Equation

What is the solution to:

\(\displaystyle \frac{x^2-6x+8}{x-2}=0\)

 

Possible Answers:

4

0

1

2

6

Correct answer:

4

Explanation:

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

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

 

Example Question #4 : How To Factor An Equation

What is the value of \(\displaystyle x\) where:

\(\displaystyle 3x+5=6-2x\)

Possible Answers:

\(\displaystyle 0\)

\(\displaystyle -1\)

\(\displaystyle 1\)

\(\displaystyle \frac{1}{5}\)

\(\displaystyle -\frac{1}{5}\)

Correct answer:

\(\displaystyle \frac{1}{5}\)

Explanation:

The question asks us to find the value of \(\displaystyle x\), 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 \(\displaystyle x\) containing numbers on the other side.

 

We utilize opposite operations to both sides by adding \(\displaystyle 2x\) to each side of the equation and get \(\displaystyle 5x+5=6\)

 

Next, we subtract \(\displaystyle 5\) from both sides, yielding

 

\(\displaystyle 5x=1\)

 

Then we divide both sides by \(\displaystyle 5\) to get rid of that \(\displaystyle 5\) on \(\displaystyle 5x\)

 

\(\displaystyle x=\frac{1}{5}\)

Example Question #1 : Factoring Equations

Factor the following equation:

\(\displaystyle {x^3-2x^2+x}\)

Possible Answers:

\(\displaystyle x^{2}(x-1)\)

\(\displaystyle x^{2}(x-1)^{2}\)

\(\displaystyle (x-1)^{3}\)

\(\displaystyle x(x-1)(x-1)\)

\(\displaystyle (x-1)^{2}\)

Correct answer:

\(\displaystyle x(x-1)(x-1)\)

Explanation:

First we factor out an x then we can factor the \(\displaystyle (x-1)(x-1)\)

Example Question #203 : Algebra

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

\(\displaystyle 4y^{2}=169x^{2}-81\)

Possible Answers:

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

\(\displaystyle \frac{(2y)^{2}}{13x+9}=27x-9-14x\)

\(\displaystyle 6y^{2}=\frac{3\times (13x+9)\times (13x-9)}{2}\)

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

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

Correct answer:

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

Explanation:

The equation presented in the problem is:

\(\displaystyle 4y^{2}=169x^{2}-81\)

We know that:

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

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

 

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

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

Multiplying everything by \(\displaystyle 13x+9\), we get back to:

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

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

 

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

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

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

 

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

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

\(\displaystyle 4y^{2}=\frac{338x^{2}-162}{2}=169x^{2}-81\)

 

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

because

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

\(\displaystyle 4y^{2}=169x^{2}-81-234x \neq 4y^{2}=169x^{2}-81\)

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