# Factoring Quadratic Equations

Factoring (also called factorising) is when terms that need to be multiplied together to get a mathematical expression are determined. For example, have a look at the quadratic expression below:

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${x}^{2}-16=\left(x-4\right)\left(x+4\right)$

In this example ${x}^{2}-16$ has been factored as we determined the terms to multiply together to get this expression: $\left(x+4\right)\left(x-4\right)$.

Factoring is a method that can be used to solve Quadratic Equations. Let's look at how we do this by continuing the example above:

${x}^{2}-16=\left(x-4\right)\left(x+4\right)=0\phantom{\rule{0ex}{0ex}}{x}_{1}=4and{x}_{2}=-4$

Determining the value of these x-intercepts is what solves the equation. They are the roots of the equation, which is when the equation = 0.

## How do we factor quadratic equations?

We can factor Quadratic Equations in one of the following ways:

### Taking the greatest common factor (GCF)

Taking the greatest common factor is when we determine the Highest Common Factor that evenly divides into all the other terms. To master this factoring method, you first need to understand the distributive property, which is when we solve expressions in the form of a(b+c) into ab+ac. For example, have a look at how this method is used with the quadratic expression below:

$8{x}^{2}\left(4x+5\right)=8{x}^{2}·4x+8{x}^{2}·5\phantom{\rule{0ex}{0ex}}=32{x}^{3}+40{x}^{2}$

Now that we've had a look at distributive properties, let us now use the example and steps below to see how we can factorise by taking the greatest common factor:

$12{x}^{2}+8x=0$

Step 1: Find the greatest common factor by identifying the numbers and variables that each term has in common.

$12{x}^{2}=4·3·x·x\phantom{\rule{0ex}{0ex}}8x=4·2·x$The Number and variables that appear the most are 4 and x, therefore making our GCF=4x. Step 2: Write out each term as a product of the greatest common factor and another factor, i.e. the two parts of the term. You can determine the other factor by dividing your term by your GCF. $\frac{12{x}^{2}}{4x}=3x\phantom{\rule{0ex}{0ex}}\therefore 12{x}^{2}=4x·3x\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\frac{8x}{4x}=2\phantom{\rule{0ex}{0ex}}\therefore 8x=4x·2$Step 3: Having rewritten your terms, rewrite your quadratic equation in the following form: $ab+ac=0\phantom{\rule{0ex}{0ex}}12{x}^{2}+8x=4x\left(3x\right)+4x\left(2\right)=0$Step 4: Apply the law of distributive property and factor out your greatest common factor. $4x\left(3x\right)+4x\left(2\right)=4x\left(3x+2\right)=0$Step 5 (solving the quadratic equation): Equate the factored expression to 0 and solve for the x-intercepts. ${x}_{1}:4x=0\phantom{\rule{0ex}{0ex}}{x}_{1}=0\phantom{\rule{0ex}{0ex}}{x}_{2}:3x+2=0\phantom{\rule{0ex}{0ex}}{x}_{2}=\frac{-2}{3}$

Factoring by taking out Common Factors can also be used. This method is similar to grouping to solve quadratic Equations, with a leading coefficient equal to 1.

${x}^{2}-6x+8=0$

Step 1: List out the values of a, b and c.

$a=1b=-6c=8$

Step 2: Find two numbers that product the constant (c) and add up to the x-coefficient (-6).

$1×8=8\phantom{\rule{0ex}{0ex}}2×4=8\phantom{\rule{0ex}{0ex}}-2×-4=8$The two numbers are -2 and -4, as they can be used to add to -6, i.e. by having -2 and -4. 1 and 8 can not be arranged in any way that would make them equal to 8. Step 3: Subtract these numbers from x and form two binomial Factors. If they were -2 and +4, for example, you would subtract 2 from and add 4 to x. ${x}^{2}-6x+8=\left(x-2\right)\left(x-4\right)$Step 4 (solving the quadratic equation): Equate the binomial Factors to 0 and solve for the x-intercepts. ${x}_{1}:x-2=0\phantom{\rule{0ex}{0ex}}x=2\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{x}_{2}:x-4=0\phantom{\rule{0ex}{0ex}}x=4$

### Perfect square method

Using the perfect square method to factorise is when we transform a perfect square trinomial ${a}^{2}+2ab+{b}^{2}$ or ${a}^{2}-2ab+{b}^{2}$into a perfect square binomial, ${\left(a+b\right)}^{2}or{\left(a-b\right)}^{2}$. All perfect square trinomials with one variable have one root.

${x}^{2}+14x+49$ is a perfect square trinomial which would be transformed into the perfect square binomial of ${\left(x+7\right)}^{2}$ . Your root in this trinomial will be x=-7. The graph of this trinomial would look like this:

Parabola of a perfect square trinomial, Nicole Moyo-StudySmarter Originals

Let's look at how to implement the perfect square method:

${x}^{2}-10x+25=0\phantom{\rule{0ex}{0ex}}$

Step 1: Transform your equation from standard form $a{x}^{2}-bx+c=0$ into a perfect square trinomial ${a}^{2}-2ab+{b}^{2}$.

${x}^{2}-10x+25={x}^{2}-2\left(x\right)\left(5\right)+{5}^{2}$

Step 2: Transform the perfect square trinomial into a perfect square binomial, ${\left(a-b\right)}^{2}$.

${x}^{2}-2\left(x\right)\left(5\right)+{5}^{2}={\left(x-5\right)}^{2}$

Step 3 (solving the quadratic equation): Calculate the value of the x-intercept by equating the perfect square binomial to 0 and solving for x.

${\left(x-5\right)}^{2}=0\phantom{\rule{0ex}{0ex}}\sqrt{{\left(x-5\right)}^{2}}=±\sqrt{0}\phantom{\rule{0ex}{0ex}}x-5=0\phantom{\rule{0ex}{0ex}}x=5$As you can see, it has one root and would be graphed like this:

Solution graph of a perfect square trinomial, Nicole Moyo-StudySmarter Originals

### Grouping

Grouping is when we group terms that have a common factor before factoring. This method is commonly used to factor quadratic Equations with a leading coefficient (a) greater than 1. Grouping can be done by following the steps below:

$3{x}^{2}+10x-8$

Step 1: List out the values of a, b and c.

$a=3b=10c=-8$

Step 2: Find the two numbers such that their product is equal to ac and the sum is equal to b.

$ac=-24b=10\phantom{\rule{0ex}{0ex}}1·24=24\phantom{\rule{0ex}{0ex}}2·12=24-2+12=10$The two numbers are -2 and 12, as they can be used to add to 10, i.e. by having -2 and +12. 1 and 24 cannot be arranged in any way that would make them equal to 10.

Step 3: Use these factors to separate the x-term (bx) in the original expression/equation.

$3{x}^{2}+10x-8=3{x}^{2}-2x+12x-8$

Step 4: Use grouping to factor the expression.

$\left(3{x}^{2}-2x\right)+\left(12x-8\right)=x\left(3x-2\right)+4\left(3x-2\right)\phantom{\rule{0ex}{0ex}}=\left(x+4\right)\left(3x-2\right)$

Step 5 (how to solve the quadratic equation): Equate the factored expression to 0 and solve for x.

$\left(x+4\right)\left(3x-2\right)=0\phantom{\rule{0ex}{0ex}}{x}_{1}:x+4=0\phantom{\rule{0ex}{0ex}}x=-4\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{x}_{2}:3x-2=0\phantom{\rule{0ex}{0ex}}x=\frac{2}{3}$

## Factoring Quadratic Equations - Key takeaways

• Factoring is when we determine which terms need to be multiplied together to get a mathematical expression.

• Taking the greatest common factor is a method of factoring where we determine the Highest Common Factor that evenly divides into all the other terms.

• Using the perfect square method is another way to factorise, where we transform a perfect square trinomial ${a}^{2}+2ab+{b}^{2}or{a}^{2}-2ab+{b}^{2}$ into a perfect square binomial, ${\left(a+b\right)}^{2}or{\left(a-b\right)}^{2}$.

• Grouping is also a method of factoring where we group terms that have a common factor before factoring.

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How do we factor Quadratic Equations?

Quadratic equations can be factored by using one of the following methods:

• Taking the greatest common factor which is when we determine the highest common factor that evenly divides into all the other terms.
• Using the perfect square method, where we transform a perfect square trinomial, a²+2ab+b² or a²-2ab+b² into a perfect square binomial, (a+b)² or (a - b)² .
• Grouping, which is when we group terms that have a common factor before factoring.

How do you solve equations by factoring?

An equation can be solved by expressing it as a product of multiple factors that equate to zero. Subsequently, the roots of the equation can be found by solving the equation "factor = 0" for each individual factor.

How do you factor quadratic equations with a coefficient of 1?

Quadratic equations with a coefficient of 1 can be factored by using the grouping method.

How do we factor equations by grouping?

2x²-8x+6

Step 1: List out the values of a, b and c.

a=2 b=-8 c=6

Step 2: Find the two numbers that product ac and also add to b.

ac=12b=-8

1x12 =12

2x6=12

-2-6=-8

The two numbers are therefore -2 and -6, as they can be used to add to -8, ie: by having -2 and -6. 1 and 12 cannot be arranged in any way that would make them equal to -8.

Step 3: Use these factors to separate the x-term (bx) in the original expression/equation.

2x²-8x+6=2x²-2x-6x+6

Step 4: Use grouping to factor the expression.

(2x²-2x)-(6x-6)=2x(x-1)-6(x-1)

=(2x-6)(x-1)

Step 5 (how to solve the quadratic equation): Equate the factored expression to 0 and solve for the x-intercepts.

(2x-6)(x-1)=0

x1:2x-6=0

x=3

x2:x-1=0

x=1

How do we factor fractions in quadratic equations?

Factoring quadratic equations with fractions is done by multiplying each term in the equation with the lowest common denominator. Let us have a look at this:

x²+1=(11/6)x-2/3

Step 1:  Multiply each term with the lowest common denominator (LCD).

In this example, LCD=6.

6(x²+1)=6((11/6)x-2/3)

6x²+6=11x-4

Step 2: Equate your equation to 0, if it hasn't already been and then factor it. We will factor our equation by using the grouping method.

6x²-11x+10=0

This quadratic equation can be solved by using formula.

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