Solving Equations by Factoring Practice Problems with Solutions

Master solving quadratic equations by factoring with step-by-step practice problems. Learn zero product property, trinomial factoring, and common factor methods.

📚Practice Solving Equations by Factoring - Build Your Skills

Apply the zero product property to solve factored equations
Factor quadratic equations using common factor extraction methods
Solve trinomial equations by recognizing perfect square patterns
Master transposing terms to set equations equal to zero
Practice identifying appropriate factoring methods for different equation types
Verify solutions by substituting back into original equations

Understanding Solving Equations by Factoring

Complete explanation with examples

To solve equations through factorization, we must transpose all the elements to one side of the equation and leave $0$ on the other side.
Why? Because after factoring, we will have $0$ as the product.

Let's remember the following property

The product of two numbers equals $0$ when, at least, one of them is $0$ .
If $x\times y=0$
then
either: $x=0$
or: $y=0$
or both are equal to $0$ .

Steps to carry out to solve equations through factorization

Let's move all the elements to one side of the equation and leave $0$ on the other.
Let's factor using one of the methods we have learned: by taking out the common factor, with shortcut multiplication formulas, or with trinomials.
Let's find out when the elements achieve a product equivalent to $0$ .

Detailed explanation

Practice Solving Equations by Factoring

Test your knowledge with 3 quizzes

In front of you is a square.

The expressions listed next to the sides describe their length.

( $$ x>-2 $$ length measurements in cm).

Since the area of the square is 16.

Find the lengths of the sides of the square.

Examples with solutions for Solving Equations by Factoring

Step-by-step solutions included

Exercise #1

Solve for x.

$-x^2-7x-12=0$

Step-by-Step Solution

First, factor using trinomials and remember that there might be more than one solution for the value of $x$ :

$-x^2-7x-12=0$

Divide by -1:

$x^2+7x+12=0$

$(x+3)(x+4)=0$

Therefore:

$x+4=0$

$x=-4$

Or:

$x+3=0$

$x=-3$

Answer:

$x=-3,x=-4$

Video Solution

Exercise #2

Find the value of the parameter x.

$(x-5)^2=0$

Step-by-Step Solution

We will factor using the shortened multiplication formulas:

$a^2-b^2=(a-b)(a+b)$ $(a-b)^2=a^2-2ab+b^2$

$(a+b)^2=a^2+2ab+b^2$

Let's remember that there might be more than one solution for the value of x.

According to one solution, we'll take the square root:

$(x-5)^2=0$

$x-5=0$

$x=5$

According to the second solution, we'll use the shortened multiplication formula:

$(x-5)^2=x^2-10x+25=0$

We'll use the trinomial:

$(x-5)(x-5)=0$

$x-5=0$

$x=5$

$x-5=0$

$x=5$

Therefore, according to all calculations, $x=5$

Answer:

$x=5$

Video Solution

Exercise #3

Find the value of the parameter x.

$x^2-25=0$

Step-by-Step Solution

We will factor using the shortened multiplication formulas:

$a^2-b^2=(a-b)(a+b)$ $(a-b)^2=a^2-2ab+b^2$

$(a+b)^2=a^2+2ab+b^2$

Remember that there might be more than one solution for the value of x.

According to the first formula:

$x^2=a^2$

We'll take the square root:

$x=a$

$25=b^2$

We'll take the square root:

$b=5$

We'll use the first shortened multiplication formula:

$a^2-b^2=(a-b)(a+b)$

$x^2-25=(x-5)(x+5)=0$

Therefore:

$x+5=0$

$x=-5$

Or:

$x-5=0$

$x=5$

Answer:

$x=5,x=-5$

Video Solution

Exercise #4

Find the value of the parameter x.

$2x^2-7x+5=0$

Step-by-Step Solution

We will factor using trinomials, remembering that there is more than one solution for the value of X:

$2x^2-7x+5=0$

We will factor -7X into two numbers whose product is 10:

$2x^2-5x-2x+5=0$

We will factor out a common factor:

$2x(x-1)-5(x-1)=0$

$(2x-5)(x-1)=0$

Therefore:

$x-1=0$ $x=1$

Or:

$2x-5=0$

$2x=5$

$x=2.5$

Answer:

$x=1,x=2.5$

Video Solution

Exercise #5

Find the value of the parameter x.

$(x-4)^2+x(x-12)=16$

Step-by-Step Solution

Let's open the parentheses, remembering that there might be more than one solution for the value of X:

$(x-4)^2+x(x-12)=16$

$x^2-8x+16+x^2-12x=16$

$2x^2-20x=0$

$2x(x-10)=0$

Therefore:

$x-10=0$

$x=10$

Or:

$2x=0$

$x=0$

Answer:

$x=0,x=10$

Video Solution

Frequently Asked Questions

Everything you need to know about Solving Equations by Factoring

What is the zero product property and why is it important for solving equations by factoring?

The zero product property states that if the product of two numbers equals zero, then at least one of them must be zero. This property is crucial because after factoring an equation, we can set each factor equal to zero to find the solutions.

What are the main steps to solve equations by factoring?

Follow these three steps: 1) Move all terms to one side leaving zero on the other, 2) Factor the expression using common factors, trinomial patterns, or shortcut formulas, 3) Apply the zero product property by setting each factor equal to zero and solve.

How do I know which factoring method to use when solving equations?

Look for these patterns: common factors first, then perfect square trinomials (a² ± 2ab + b²), difference of squares (a² - b²), or general trinomials (ax² + bx + c). Start with the simplest method and work toward more complex ones.

Why must I set the equation equal to zero before factoring?

Setting the equation equal to zero allows you to use the zero product property after factoring. Without zero on one side, you cannot conclude that individual factors equal zero, making it impossible to find the solutions.

What should I do if my factored equation has repeated factors like (x-7)(x-7)?

When you have repeated factors, the solution occurs at the value that makes the factor zero. In (x-7)(x-7) = 0, both factors give x = 7, so there is one solution with multiplicity 2.

How can I check if my solutions are correct after solving by factoring?

Substitute each solution back into the original equation. If both sides are equal, your solution is correct. For example, if x = 7 is your solution to x² + 49 = 14x, check: 7² + 49 = 14(7), which gives 98 = 98 ✓

What are common mistakes students make when solving equations by factoring?

Common errors include: forgetting to move all terms to one side, factoring incorrectly, not applying the zero product property to all factors, and arithmetic mistakes when solving simple equations like x - 7 = 0.

Can all quadratic equations be solved by factoring?

No, not all quadratic equations can be factored using integers. Some equations require the quadratic formula or completing the square. However, many common quadratic equations in algebra courses are designed to factor nicely.

Continue Your Math Journey

Master Solving Equations by Factoring first, then advance to these related topics that build on your fraction skills

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Solving Equations by Factoring Practice Problems with Solutions

Understanding Solving Equations by Factoring

Let's remember the following property

Steps to carry out to solve equations through factorization

Practice Solving Equations by Factoring

Examples with solutions for Solving Equations by Factoring

Frequently Asked Questions

What is the zero product property and why is it important for solving equations by factoring?

What are the main steps to solve equations by factoring?

How do I know which factoring method to use when solving equations?

Why must I set the equation equal to zero before factoring?

What should I do if my factored equation has repeated factors like (x-7)(x-7)?

How can I check if my solutions are correct after solving by factoring?

What are common mistakes students make when solving equations by factoring?

Can all quadratic equations be solved by factoring?

Continue Your Math Journey

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Topics Learned in Later Sections

Practice by Question Type