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Factor Quadratics
to Find Zeros

Learn how to factor quadratic expressions and use those factors to reveal the zeros of the function they define.

Algebra 2

What Is a Quadratic Expression?

A quadratic expression is a polynomial of degree 2. Its general form is:

\(a\): Leading Coefficient
Controls the width and direction (up or down) of the parabola.
\(b\): Linear Coefficient
Shifts the axis of symmetry left or right.
\(c\): Constant Term
The \(y\)-intercept: where the parabola crosses the \(y\)-axis.

What Are the Zeros of a Function?

The zeros (also called roots or x-intercepts) of a function \(f(x)\) are the values of \(x\) where the graph crosses the \(x\)-axis: that is, where \(f(x) = 0\).

x y 1 2 3 4 1 2
\(x = 1\)
\(x = 4\)
vertex

Why Does Factoring Matter?

Factoring is the bridge between the standard form of a quadratic and its zeros. It converts one equation into a product of simpler expressions, making the zeros easy to read off.

Standard Form
\(x^2 - 5x + 4\)
Zeros are hidden.
Factored Form
\((x - 1)(x - 4)\)
Zeros are visible: \(x = 1\) and \(x = 4\).

The Zero Product Property

This property is the foundation of solving quadratics by factoring. Once a quadratic is written as a product of two linear factors, each factor is set equal to zero, yielding the zeros of the original function.

1
Factor the quadratic: \(ax^2 + bx + c = (px + q)(rx + s)\).
2
Set each factor to zero: \(px + q = 0\) and \(rx + s = 0\).
3
Solve each equation to find the zeros.

Factoring When \(a = 1\)

When the leading coefficient is 1, the quadratic has the form \(x^2 + bx + c\). To factor, find two numbers \(p\) and \(q\) such that:

Sum
\(p + q = b\)
Product
\(p \cdot q = c\)

Zeros vs. Factors: Key Distinction

Students often confuse the numbers in the factors with the zeros. They are related but have opposite signs.

Factored Form Constants in Factors Zeros
\((x - 3)(x - 7)\) \(-3\) and \(-7\) \(x = 3\) and \(x = 7\)
\((x + 2)(x - 5)\) \(+2\) and \(-5\) \(x = -2\) and \(x = 5\)
\((x + 4)(x + 1)\) \(+4\) and \(+1\) \(x = -4\) and \(x = -1\)

How to Factor and Find Zeros

1
Write in standard form: arrange as \(ax^2 + bx + c = 0\).
2
Factor out GCF (greatest common factor) if one exists.
3
Find two numbers whose product is \(a \cdot c\) and whose sum is \(b\).
4
Split the middle term using those numbers and factor by grouping (if \(a \neq 1\)).
5
Set each factor equal to zero and solve for \(x\).

Example 1: \(x^2 - 5x + 6\)

Find the zeros of \(f(x) = x^2 - 5x + 6\).

1
Need two numbers with product = \(6\) and sum = \(-5\).
2
The numbers are \(-2\) and \(-3\), since \((-2)(-3) = 6\) and \((-2)+(-3) = -5\).
3
Factor: \(x^2 - 5x + 6 = (x - 2)(x - 3)\).
4
Set each factor to zero: \(x - 2 = 0 \Rightarrow\) \(x = 2\) and \(x - 3 = 0 \Rightarrow\) \(x = 3\).

Graph of \(f(x) = x^2 - 5x + 6\)

x y 1 2 3 4 5 1 2 3 4 5 6
\(x = 2\)
\(x = 3\)
\((0, 6)\)
vertex \((2.5,\, -0.25)\)

Example 2: \(x^2 + 2x - 15\)

Find the zeros of \(f(x) = x^2 + 2x - 15\).

1
Need two numbers with product = \(-15\) and sum = \(2\).
2
The numbers are \(5\) and \(-3\), since \(5 \cdot (-3) = -15\) and \(5 + (-3) = 2\).
3
Factor: \(x^2 + 2x - 15 = (x + 5)(x - 3)\).
4
Zeros: \(x + 5 = 0 \Rightarrow\) \(x = -5\) and \(x - 3 = 0 \Rightarrow\) \(x = 3\).

Example 3: \(2x^2 + 7x + 3\)

Find the zeros of \(f(x) = 2x^2 + 7x + 3\). Here \(a = 2 \neq 1\), so we use the AC method.

1
Compute \(a \cdot c = 2 \cdot 3 = 6\). Find two numbers with product = \(6\) and sum = \(7\).
2
The numbers are \(1\) and \(6\).
3
Split the middle term: \(2x^2 + x + 6x + 3\).
4
Factor by grouping: \(x(2x + 1) + 3(2x + 1) = (2x + 1)(x + 3)\).
5
Zeros: \(2x + 1 = 0 \Rightarrow\) \(x = -\tfrac{1}{2}\) and \(x + 3 = 0 \Rightarrow\) \(x = -3\).

Always Check for a GCF First

Before attempting to factor the trinomial, always check whether all terms share a greatest common factor.

The GCF of \(3\) does not change the zeros. Setting \(3(x-1)(x-3) = 0\):

Zero 1
\(x - 1 = 0\), so \(x = 1\).
Zero 2
\(x - 3 = 0\), so \(x = 3\).

Special Factoring Patterns

Difference of Squares
\(a^2 - b^2 = (a+b)(a-b)\)

Example: \(x^2 - 9 = (x+3)(x-3)\)
Zeros: \(x = -3\) and \(x = 3\).
Perfect Square Trinomial
\(a^2 \pm 2ab + b^2 = (a \pm b)^2\)

Example: \(x^2 - 6x + 9 = (x-3)^2\)
Repeated zero: \(x = 3\).

Common Mistakes to Avoid

✕ Sign Errors in Zeros
Seeing \((x + 3)\) and writing \(x = 3\) instead of \(x = -3\). Always solve \(x + 3 = 0\).
✕ Forgetting the GCF
Trying to factor \(4x^2 - 8x\) as a trinomial instead of first pulling out \(4x\).
✕ Dividing by \(x\)
Dividing both sides of \(x^2 - 5x = 0\) by \(x\), losing the zero at \(x = 0\). Factor instead: \(x(x-5) = 0\).

Factoring Methods Summary

Method When to Use Example
GCF First All terms share a factor \(6x^2 - 12x = 6x(x - 2)\)
Simple Trinomial \(a = 1\): find sum/product pair \(x^2 - 7x + 10 = (x-2)(x-5)\)
AC Method \(a \neq 1\): split middle term \(2x^2 + 5x + 3 = (2x+3)(x+1)\)
Difference of Squares \(b = 0\), two perfect squares \(x^2 - 25 = (x+5)(x-5)\)
Perfect Square Fits the pattern \(a^2 \pm 2ab + b^2\) \(x^2 + 10x + 25 = (x+5)^2\)

Question 1 of 5

What are the zeros of \(f(x) = x^2 - 9x + 20\)?

Question 2 of 5

Which is the factored form of \(x^2 + 3x - 28\)?

Question 3 of 5

What are the zeros of \(f(x) = 3x^2 - 12x\)?

Question 4 of 5

How many distinct zeros does \(f(x) = x^2 - 10x + 25\) have?

Question 5 of 5

What are the zeros of \(f(x) = 2x^2 + 5x - 3\)?