Factoring Trinomials Practice Problems and Step-by-Step Solutions

Master factoring trinomials with guided practice problems using AC method and quadratic formula. Build confidence solving ax²+bx+c expressions step-by-step.

📚Master Factoring Trinomials with Interactive Practice

Factor trinomials using the AC method to find number pairs
Apply the quadratic formula to solve ax²+bx+c expressions
Identify coefficients a, b, and c in trinomial expressions
Convert trinomials into factored form (x+p)(x+q)
Solve real-world problems involving trinomial factorization
Handle both positive and negative coefficient scenarios

Understanding Factoring Trinomials

Complete explanation with examples

I present to you the following trinomial

$ax^2+bx+c$

Regardless of whether the coefficients of the terms are positive or negative, as long as they appear in the style of a trinomial, the exercise will be called "trinomial".

The factorization will look like this:

$(x+solution \space one)(x+solution\space two)$
or with subtractions, depending on the solutions.

Detailed explanation

Practice Factoring Trinomials

Test your knowledge with 17 quizzes

Solve the following problem:

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

Examples with solutions for Factoring Trinomials

Step-by-step solutions included

Exercise #1

$x^2+10x+16=0$

Step-by-Step Solution

Let's observe that the given equation:

$x^2+10x+16=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+10x+16=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=16\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+2)(x+8)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+2)(x+8)=0 \\ \downarrow\\ x+2=0\rightarrow\boxed{x=-2}\\ x+8=0\rightarrow\boxed{x=-8}\\ \boxed{x=-2,-8}$ Therefore, the correct answer is answer B.

Answer:

$x=-8,x=-2$

Video Solution

Exercise #2

$x^2+10x-24=0$

Step-by-Step Solution

Let's observe that the given equation:

$x^2+10x-24=0$ is a quadratic equation that can be solved using quick factoring:

$x^2+10x-24=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-24\\ \underline{?}+\underline{?}=10\end{cases}\\ \downarrow\\ (x+12)(x-2)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x+12)(x-2)=0 \\ \downarrow\\ x+12=0\rightarrow\boxed{x=-12}\\ x-2=0\rightarrow\boxed{x=2}\\ \boxed{x=-12,2}$ Therefore, the correct answer is answer B.

Answer:

$x=2,x=-12$

Video Solution

Exercise #3

$x^2-19x+60=0$

Step-by-Step Solution

Let's observe that the given equation:

$x^2-19x+60=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-19x+60=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=60\\ \underline{?}+\underline{?}=-19\end{cases}\\ \downarrow\\ (x-4)(x-15)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-4)(x-15)=0 \\ \downarrow\\ x-4=0\rightarrow\boxed{x=4}\\ x-15=0\rightarrow\boxed{x=15}\\ \boxed{x=4,15}$ Therefore, the correct answer is answer A.

Answer:

$x=15,x=4$

Video Solution

Exercise #4

$x^2-2x-3=0$

Step-by-Step Solution

Let's observe that the given equation:

$x^2-2x-3=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-2x-3=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-3\\ \underline{?}+\underline{?}=-2\end{cases}\\ \downarrow\\ (x-3)(x+1)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-3)(x+1)=0 \\ \downarrow\\ x-3=0\rightarrow\boxed{x=3}\\ x+1=0\rightarrow\boxed{x=-1}\\ \boxed{x=-1,3}$ Therefore, the correct answer is answer B.

Answer:

$x=3,x=-1$

Video Solution

Exercise #5

$x^2-3x-18=0$

Step-by-Step Solution

Let's observe that the given equation:

$x^2-3x-18=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-3x-18=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-18\\ \underline{?}+\underline{?}=-3\end{cases}\\ \downarrow\\ (x-6)(x+3)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-6)(x+3)=0 \\ \downarrow\\ x-6=0\rightarrow\boxed{x=6}\\ x+3=0\rightarrow\boxed{x=-3}\\ \boxed{x=6,-3}$ Therefore, the correct answer is answer A.

Answer:

$x=-3,x=6$

Video Solution

Frequently Asked Questions

Everything you need to know about Factoring Trinomials

What is a trinomial in algebra?

A trinomial is an algebraic expression with three terms in the form ax²+bx+c, where 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term. The coefficient 'a' cannot be zero, but it can be positive or negative.

How do you factor trinomials using the AC method?

The AC method involves finding two numbers whose product equals a×c and whose sum equals b. First, list all factor pairs of a×c, then identify which pair adds up to b. These numbers become your solutions for factoring the trinomial into (x+solution₁)(x+solution₂).

When should I use the quadratic formula for factoring trinomials?

Use the quadratic formula x = (-b ± √(b²-4ac))/2a when the AC method becomes difficult or when you need exact decimal solutions. This method is particularly useful for trinomials that don't factor nicely into whole numbers.

What are the steps to factor x²+8x+12?

Step 1: Find factor pairs of 12: (1,12), (2,6), (3,4). Step 2: Check which pair adds to 8: 2+6=8. Step 3: Write the factored form: (x+2)(x+6). Step 4: Verify by expanding: (x+2)(x+6) = x²+8x+12 ✓

How do I handle negative coefficients in trinomial factoring?

When coefficients are negative, consider both positive and negative factor pairs of a×c. If solutions are negative, subtract them from x instead of adding: (x-solution₁)(x-solution₂). Always check your signs carefully in the final factored form.

What does it mean when a trinomial has the same solution twice?

When both solutions are identical (like -2 and -2), the trinomial is a perfect square. This results in factorization like (x-2)(x-2) or (x-2)². The discriminant (b²-4ac) equals zero in these cases.

Why is factoring trinomials important in algebra?

Factoring trinomials is essential for: solving quadratic equations, simplifying algebraic fractions, finding x-intercepts of parabolas, and solving real-world problems involving area, projectile motion, and optimization. It's a foundational skill for advanced algebra topics.

What common mistakes should I avoid when factoring trinomials?

Common mistakes include: forgetting to consider negative factor pairs, mixing up the signs in the final answer, not checking if 'a' equals 1 before applying methods, and failing to verify answers by expanding the factored form back to the original trinomial.

Continue Your Math Journey

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

Topics Learned in Later Sections

Algebraic Fractions Simplifying Algebraic Fractions Factoring Algebraic Fractions Addition and Subtraction of Algebraic Fractions Multiplication and Division of Algebraic Fractions Solving Equations by Factoring Uses of Factorization

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Applying the formula Complete the equation Equation factorization How many solutions exist to a problem? More than one factorization Solving the equation Solving the problem Test if the coefficient is different from 1 True / false Using quadrilaterals Using short multiplication formulas Worded problems

Factoring Trinomials Practice Problems and Step-by-Step Solutions

Understanding Factoring Trinomials

I present to you the following trinomial

$ax^2+bx+c$

The factorization will look like this:

Practice Factoring Trinomials

Examples with solutions for Factoring Trinomials

Frequently Asked Questions

What is a trinomial in algebra?

How do you factor trinomials using the AC method?

When should I use the quadratic formula for factoring trinomials?

What are the steps to factor x²+8x+12?

How do I handle negative coefficients in trinomial factoring?

What does it mean when a trinomial has the same solution twice?

Why is factoring trinomials important in algebra?

What common mistakes should I avoid when factoring trinomials?

More Factoring Trinomials Questions