Factored Quadratics: Roots and x-Intercepts

Ontario Grade 11 Mathematics

Quadratic Functions

How a factored equation reveals where a graph meets the x-axis

A graph’s x-intercepts are the points where it meets or touches the x-axis. A factored equation can show these points directly. In this lesson, you will connect the factors of a quadratic function to the x-values of its x-intercepts. The main idea is simple: at an x-intercept, the output is zero. A factor that becomes zero makes the whole product zero.

What you will learn

Recognize an x-intercept on a graph and in an equation.
Explain how a factor can identify an x-intercept.
Find the x-intercepts of a factored quadratic function.
Check an x-intercept by substituting its x-value into the function.

1. Prerequisite bridge: coordinates and zero products

A point on a graph has an x-coordinate and a y-coordinate. For a function written as

y=f(x)

, the value of

f(x)

is the y-coordinate produced by an input

x

.

The x-axis is the horizontal axis. Every point on it has a y-coordinate of zero. Therefore, an x-intercept has the form

(r,0)

. The number

r

is the x-value where the graph meets the axis.

A factor is an expression that is multiplied by another expression. For example, in

y=(x-3)(x+2)

, the expressions

(x-3)

and

(x+2)

are factors. A product is zero if at least one of its factors is zero. This is the key fact that links factors to x-intercepts.

(x-a)(x-b)=0

An x-intercept has a y-coordinate of zero.
At an x-intercept, set the function’s output equal to zero.
If one factor in a product is zero, the entire product is zero.

2. The connection in plain language

Consider the function

y=(x-3)(x+2)

. If

x=3

, then the first factor is zero. The product is zero, so the point

(3,0)

is an x-intercept.

If

x=-2

, then the second factor is zero. The product is again zero, so

(-2,0)

is also an x-intercept. The signs inside the factors can feel surprising: the factor

(x+2)

is zero when

x=-2

.

This gives a useful rule for a function in factored form. Set each factor equal to zero and solve for

x

. Each solution gives an x-value where the function has output zero. Write the intercept as a point by pairing that x-value with zero.

This rule identifies x-intercepts, not every point on the graph. For other points, choose an x-value and calculate the function output.

(x-a)=0\Rightarrow x=a

For a factored function, set each factor equal to zero.
The solutions are x-values; the matching intercepts are points with y-coordinate zero.
Use the factor itself to determine the sign of the x-value.

3. Read factors, graph points, and symbols together

A factored equation, a coordinate pair, and a graph describe the same intercept in different ways. The equation shows which factor becomes zero. The coordinate pair records both coordinates. The graph shows where the curve meets the horizontal axis.

For example, when a factor gives

x=3

, the corresponding graph point is

(3,0)

. On a sketch, this point lies on the x-axis. The other coordinate must be zero because every point on that axis has a zero y-coordinate.

A quadratic in factored form may have two different linear factors and therefore two different x-intercepts. It may also have a repeated factor, such as

(x-4)^2

. That factor is zero at

x=4

, so the graph has the x-intercept

(4,0)

. The factor gives the location; the equation alone does not require you to draw the whole graph to identify it.

If a quadratic is not already written as a product of factors, this method cannot be read directly from the equation. In this lesson, focus on connecting factors that are already visible with the intercepts they identify.

(x-r)^2=0\Rightarrow (r,0)

A factor’s zero gives an x-value, while the x-intercept is written as a point.
A repeated factor still identifies its zero and corresponding x-intercept.
The factored form makes the intercept locations visible.

4. Guided example and application

Suppose a quadratic model is given in factored form. To find where its graph meets the x-axis, use the condition that the output is zero. Then use the factors to find the x-values. The example shows each step and checks the result in the original function.

The same reasoning can help interpret a graph in context. If the horizontal axis represents time and the vertical axis represents height above a reference level, an x-intercept marks a time when the height is zero. The factor that becomes zero identifies that time, provided the model’s context assigns those meanings to its axes.

(x-a)(x-b)=0

Set the output to zero because x-intercepts lie on the x-axis.
Solve each factor separately.
Check by substituting each x-value into the original function.

Matching a factor with its x-intercept

Factor	Value of x that makes it zero	x-intercept
$(x-5)$	$5$	$(5,0)$
$(x+1)$	$-1$	$(-1,0)$
$(x-4)^2$	$4$	$(4,0)$

Worked example

Finding the intercepts from two factors

Find the x-intercepts of

f(x)=(x-5)(x+1)

.

Use the x-axis condition
At an x-intercept, the function’s output is zero. Set $f(x)$ equal to zero so the equation describes points on the x-axis.
$0=(x-5)(x+1)$
Find the first factor’s zero
The product is zero when at least one factor is zero. Set the first factor equal to zero and solve for its x-value.
$x-5=0\Rightarrow x=5$
Find the second factor’s zero
Repeat the same reasoning with the second factor. It becomes zero when $x$ is negative one.
$x+1=0\Rightarrow x=-1$
Write the intercepts as points
Each x-value pairs with a y-value of zero. These are the points where the graph meets the x-axis.
$(5,0),\;(-1,0)$
Check in the original function
Substitute each x-value into the original product. Each substitution makes one factor zero, confirming that the function output is zero.
$f(5)=0,\qquad f(-1)=0$

Answer: The x-intercepts are

(5,0)

and

(-1,0)

.

Check: At

x=5

, the factor

(x-5)

is zero. At

x=-1

, the factor

(x+1)

is zero. In both cases, the product and function output are zero.

Common mistakes and how to avoid them

Reading the number in a factor as the x-value without checking its sign.

Correction: Set the factor equal to zero. For example,

(x+1)=0

gives

x=-1

.

Giving an x-value as the complete x-intercept.

Correction: An x-value tells where the intercept occurs, but the intercept is a point. Pair the x-value with zero, such as

(-1,0)

.

Setting the whole function equal to a nonzero value when looking for an x-intercept.

Correction: An x-intercept lies on the x-axis, where the output is zero. Set

f(x)=0

.

Thinking a repeated factor gives two different x-intercepts.

Correction: A repeated factor such as

(x-4)^2

has the zero

x=4

. It identifies the x-intercept

(4,0)

.

Lesson summary

Every x-intercept has the form $(r,0)$ because its y-coordinate is zero.
For a function in factored form, set each factor equal to zero to find the x-values of its x-intercepts.
Write each x-value as a point by pairing it with zero.
Substitution into the original function checks that the output is zero.

Check your understanding

Question 1

What are the x-intercepts of

g(x)=(x-2)(x+6)

?

$(2,0)$ and $(-6,0)$
$(-2,0)$ and $(6,0)$
$(2,6)$ and $(-6,2)$
$(0,2)$ and $(0,-6)$

Show answer and explanation

(2,0)

and

(-6,0)

Set each factor to zero. The first gives

x=2

and the second gives

x=-6

. Pair each value with zero to write the intercepts.

Question 2

For

h(x)=(x+3)(x-7)

, which statement explains why

(-3,0)

is an x-intercept?

At $x=-3$ , the factor $(x+3)$ is zero, so the function output is zero.
At $x=-3$ , the factor $(x-7)$ is zero, so the function output is zero.
The number $-3$ is the y-coordinate of every x-intercept.
The factors must be added to get the x-intercept.

Show answer and explanation

At

x=-3

, the factor

(x+3)

is zero, so the function output is zero.

Substituting

x=-3

makes

(x+3)

equal zero. The product is therefore zero, giving the point

(-3,0)

.

Question 3

A function has a factor

(x-8)^2

. What x-intercept does this factor identify?

$(8,0)$
$(-8,0)$
$(0,8)$
There is no x-intercept because the factor is repeated.

Show answer and explanation

(8,0)

The factor is zero when

x=8

. The corresponding point on the x-axis is

(8,0)

.

Key terms

Factor: An expression that is multiplied by another expression.
Function output: The y-value produced by a function for a chosen x-value.
x-intercept: A point where a graph meets or touches the x-axis. Its y-coordinate is zero.
Factored form: A way to write an expression as a product of factors.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A1.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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A1.5 · Factored Quadratics: Roots and x-Intercepts

Ontario Grade 11 Mathematics

How a factored equation reveals where a graph meets the x-axis

What you will learn

1. Prerequisite bridge: coordinates and zero products

2. The connection in plain language

3. Read factors, graph points, and symbols together

4. Guided example and application

Matching a factor with its x-intercept

Finding the intercepts from two factors

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

Tutor support for A1.5 · Connect factors with x-intercepts

About this lesson and its review