#### Introduction

- A quadratic equation of the form ax
^{2 }+ bx + c = 0 can also be represented as

Y = ax^{2 }+ bx + c - The graph Y= ax
^{2 }+ bx + c is in the form of a parabola used to locate the solutions or roots of the quadratic equation.

## Solving Quadratic Equations by Factoring

The standard form of the quadratic equation is ax^{2 }+ bx + c = 0 which forms a parabola.

### 1. Zero-Product Property

Zero product property says that if the product of two numbers is zero, then either of the numbers or both the numbers must be equal to zero.** **

**Example 1:**

Determine the solutions of equation (x-8) (9x-4)=0 by factoring.

**Solution:**

**Step 1:** Given equation is (x-8) (9x-4)=0… (1)

**Step 2:** Now we will use zero product property for equation (1),

(x – 8) =0 or (9x – 4) =0

x = 8 and x =4/9

The solutions of the quadratic equation (x-8) (9x-4) = 0 are x = 8 and x =4/9

**Example 2:**

Determine the solutions of the quadratic equation x^{2}+ 3x-4 = 0 by factoring. Give answers to 1 decimal place where appropriate.

**Solution:**

**Step 1:** The given equation is x^{2}+ 3x-4 = 0, which is in the standard form.

**Step 2:** Now find the set of factors to factorize x^{2}+ 3x-4 = 0 … (1)

Set of factors: (1, – 4), (-1, 4), (2, -2)

The factor set (-1, 4) gives the sum as 3 and the product as -4.

So, we consider this set.

**Step 3:** Now we will rewrite the standard form into factorized form.

(x-1) (x+4) = 0 … (2)

**Step 4:** Now we will use zero product property for equation (2),

(x -1) =0 or (x+4) =0

x = 1 and x = -4

The solutions of the quadratic equation x^{2}+ 3x-4 = 0 are x = 1 and x = -4.

### 2. Plot the Graph for the Factorized Quadratic Equations

We can plot the points and draw the graph for the quadratic equations.

First, we will find the exact factors of the quadratic equations, then using these x-intercept values; we will find the vertex and then plot the graph.

Let us learn this from an example.

**Example:**

Determine the solutions of the quadratic equation x^{2}-7x + 12 = 0 by factoring and plotting the graph.

**Solution:**

**Step 1:** Given quadratic equation x^{2}-7x + 12 = 0 is in the standard form.

**Step 2:** Now find the set of factors to factorize x^{2}-7x + 12 = 0 … (1)

Set of factors: (-4, – 3), (-6, 2), (2, -6) (12, -1) (-12, 1)

The factor set (-4, -3) gives the sum as -7 and the product as 12

So, we consider this set.

**Step 3:** Now we will rewrite the standard form into factorized form.

(x – 4) (x – 3) = 0 … (2)

**Step 4:** Now we will use zero product property for equation (2),

(x – 4) =0 or (x – 3) = 0

x = 4, x = 3

The solutions of the quadratic equation x^{2}-7x + 12 = 0 are x = 4 and x = 3.

**Step 5: **Now find the coordinates of the vertex by taking the average of x-intercepts = (3+4)/2 = 3.5

Now *x* coordinate is 3.5.

Now substitute the x-coordinate in x^{2}-7x + 12 = 0.

We get, f(x) = 3.5^{2}– 7(3.5)^{2} + 12 = – 0.25

The vertex of the graph is (3.5, -0.25)

**Step 6:** Plot the vertex and x-intercepts on the graph and draw the parabola. We get,

### 3. Factored Form of Quadratic Equation by the Graph

Can we find the factored form of the quadratic equation using the graph?

Let us see this.

**Example 1:**

Given a graph, as shown in the image below, find the factored form of the quadratic equation.

**Solution:**

**Step 1:**

Now see the x-intercepts and note them from the graph.

From the graph x=3, x=8

**Step 2:**

We write the quadratic equation in factored form:

a (x-p) (x-q) = 0

Substitute x-intercepts in the equation, we get

a (x-3) (x-8) = 0

**Step 3:** Now we find the ‘a’ value by vertex (5.5, -6.25)

f (x) = a(x-3) (x-8)

-6.25 = a (5.5 -3) (5.5 -8)

-6.25 = a (2.5)(-2.5)

a = 1

So, the factorized form of the quadratic equation is 1 (x-3) (x-8) = 0.

**Example 2:**

Given a graph, as shown in the image below, find the factored form of the quadratic equation.

**Solution:**

**Step 1:** Now see the x-intercepts and note them from the graph.

From the graph x=1, x=1

**Step 2:** We write the quadratic equation in factored form:

a (x-1)(x-1) = 0

Substitute x-intercepts in the equation, we get

a (x-1)(x-1) = 0

**Step 3:** Write the quadratic equation

The factorized form of the quadratic equation is (x-1)(x-1) = 0

The quadratic equation is x^{2}– 2x + 1 = 0

**Example 3:**

Given a graph as shown in the image below, find the factored form of the quadratic equation.

**Solution:**

**Step 1:** Now see the x-intercepts and note them from the graph.

From the graph x=3, x=3

**Step 2:** We write the quadratic equation in factored form:

a (x-P) (x-q) = 0

Substitute x-intercepts in the equation, we get

a (x- 3) (x- 3) = 0

**Step 3:** Write the quadratic equation

The factorized form of the quadratic equation is (x- 3) (x- 3) = 0.

x^{2} -6x + 9 = 0 is the quadratic equation.

#### Verifying Solutions

- How can we verify whether the solution obtained is true or not?

Yes, we can verify the solution by substituting the roots in the equation.

Let us understand this from an example.

**Example:**

The solutions of the quadratic equation x^{2 }-7x + 12 = 0 by factoring are x = 3 and x = 4.

**Solution:**

Let us substitute these values in the equation x^{2 }-7x + 12 = 0… (1).

Let us take x = 3.

We get, L.H.S = (3)^{2} – 7 (3) + 12,

L.H.S = 9 -21 + 12

L.H.S = 0 = R.H.S

When x = 4

We get, L.H.S = (4)^{2} – 7 (4) + 12

L.H.S = 16 -28 + 12,

L.H.S = 0 = R.H.S

The solution is verified.

### Quadratic Equations in Real Life:

Let us learn the use of quadratic equations in our real lives.

Example:

The area of the image shown below is 196 sq.m; write the area in the form of a quadratic equation.

**Solution:**

From the image, we can write as (x+8)(x+18) = area

Given area = 196 sq. m

(x + 8) ( x+ 18) = 196

x^{2} + 8x +144 = 196

x^{2} + 26x -52 = 0

This is the quadratic equation for the given image.

#### Exercise:

- Determine the solutions of the quadratic equation x
^{2}– 10 x + 25 = 0 by factoring. - Determine the solutions of the quadratic equation by x
^{2 }– 6x + 5 + 0 factoring. - Determine the solutions of the quadratic equation x
^{2}– 9x + 9 = 0 by factoring - Convert the quadratic equation x
^{2}– 13 x + 12 = 0 into factored form. - Write the equation (x-3) (x-4) = 0 into the standard form of the quadratic equation.
- Determine the solutions of the quadratic equation 4x
^{2}– 3x + 10 = 0 by factoring. - Determine the solutions of the quadratic equation 3x
^{2}– 2x + 1 = 0 by factoring. - Find the solution of the quadratic equation x
^{2}– 7x + 12 = 0 using zero product property. - Determine the solutions of the quadratic equation x
^{2}-18x + 72 = 0 by factoring. - Determine the type of solutions for the quadratic equation x
^{2 }+ 15 x + 56 = 0by factoring.

#### Concept Map:

#### What Have We Learned:

- Quadratic equations by factoring
- Zero product property or null factor law
- A standard form of quadratic equation

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: