Solving Equations by Factoring Practice Problems

Master solving quadratic equations by factoring with step-by-step practice problems. Learn factorization techniques for zero product property equations.

📚What You'll Practice in This Section

Factor quadratic expressions like x² + 5x + 4 to solve equations
Apply the zero product property to find equation solutions
Extract common factors from expressions like 2x² + 2x = 0
Solve trinomial equations using factorization methods
Identify when factoring is the best method for solving equations
Check solutions by substituting back into original equations

Understanding Uses of Factorization

Complete explanation with examples

Factorization is the main key to solving more complex exercises than those you have studied up to today.
Factorization helps to solve different exercises, among them, those that have algebraic fractions.
In exercises where the sum or difference of their terms equals zero, factorization allows us to see them as a multiplication of $0$ and thus discover the terms that lead them to this result.

For exercises composed of fractions with expressions that may seem complicated, we can break them down into factors, reduce them, and thus end up with much simpler fractions.

Detailed explanation

Practice Uses of Factorization

Test your knowledge with 3 quizzes

In front of you is a square.

The expressions listed next to the sides describe their length.

( $$ x>-4 $$ length measurements in cm).

Since the area of the square is 36.

Find the lengths of the sides of the square.

Examples with solutions for Uses of Factorization

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.

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

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 #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.

$(x-5)^2=0$

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 #5

Find the value of the parameter x.

$(x+5)^2=0$

Step-by-Step Solution

To solve the equation $(x+5)^2 = 0$ , we will use the fact that a perfect square is zero only when the quantity being squared is zero itself.

Step 1: Set $x+5 = 0$ since it is the term being squared.
Step 2: Solve for $x$ by subtracting 5 from both sides: $x+5 = 0$ $x = -5$

Therefore, the solution to the equation is $x = -5$ .

Answer:

$x=-5$

Video Solution

Frequently Asked Questions

Everything you need to know about Uses of Factorization

What is the zero product property in factoring equations?

The zero product property states that if a product of factors equals zero, then at least one of the factors must equal zero. For example, if (x+4)(x+1) = 0, then either x+4 = 0 or x+1 = 0, giving us solutions x = -4 or x = -1.

How do you solve x² + 5x + 4 = 0 by factoring?

To solve x² + 5x + 4 = 0 by factoring: 1) Find two numbers that multiply to 4 and add to 5 (which are 4 and 1), 2) Factor as (x+4)(x+1) = 0, 3) Set each factor to zero: x+4 = 0 or x+1 = 0, 4) Solve to get x = -4 or x = -1.

When should I use factoring to solve equations?

Use factoring to solve equations when the equation equals zero and can be written as a product of simpler expressions. This method works best for quadratic equations, equations with common factors, and expressions that fit special factoring patterns like difference of squares.

What's the first step when factoring equations like 2x² + 2x = 0?

The first step is to look for a common factor in all terms. In 2x² + 2x = 0, factor out the common factor 2x to get 2x(x + 1) = 0. Then apply the zero product property: either 2x = 0 (so x = 0) or x + 1 = 0 (so x = -1).

How do I check if my factoring solutions are correct?

Substitute each solution back into the original equation to verify it makes the equation true. For example, if you found x = -1 and x = -4 for x² + 5x + 4 = 0, check: (-1)² + 5(-1) + 4 = 1 - 5 + 4 = 0 ✓ and (-4)² + 5(-4) + 4 = 16 - 20 + 4 = 0 ✓.

What are common mistakes when solving equations by factoring?

Common mistakes include: • Forgetting to set the equation equal to zero before factoring • Missing solutions when a factor appears multiple times • Arithmetic errors when finding factor pairs • Not checking solutions in the original equation • Incorrectly applying the zero product property to non-zero products.

Can all quadratic equations be solved by factoring?

No, not all quadratic equations can be easily factored using integers. Some quadratics require the quadratic formula or completing the square. However, many textbook problems are designed to factor nicely, making factoring a valuable first method to try.

How does factoring help with algebraic fractions?

Factoring helps simplify algebraic fractions by revealing common factors in numerators and denominators that can be canceled. For example, (x² - 4)/(x + 2) can be factored as (x - 2)(x + 2)/(x + 2), which simplifies to (x - 2) when x ≠ -2.

Continue Your Math Journey

Master Uses of Factorization first, then advance to these related topics that build on your fraction skills

Topics Learned in Later Sections

Solving Equations by Factoring

Practice by Question Type

Choose your preferred learning style and practice format for fraction problems

Solving the problem Using short multiplication formulas

Solving Equations by Factoring Practice Problems

Understanding Uses of Factorization

Practice Uses of Factorization

Examples with solutions for Uses of Factorization

Frequently Asked Questions

What is the zero product property in factoring equations?

How do you solve x² + 5x + 4 = 0 by factoring?

When should I use factoring to solve equations?

What's the first step when factoring equations like 2x² + 2x = 0?

How do I check if my factoring solutions are correct?

What are common mistakes when solving equations by factoring?

Can all quadratic equations be solved by factoring?

How does factoring help with algebraic fractions?

Continue Your Math Journey

Suggested Topics to Practice in Advance

Topics Learned in Later Sections

Practice by Question Type