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Factoring Quadratic Equations

- Calculus
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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:

In this example has been factored as we determined the terms to multiply together to get this expression: .

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:

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.

We can factor quadratic equations in one of the following ways:

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:

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:

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

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.

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

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

Using the perfect square method to factorise is when we transform a perfect square trinomial or into a perfect square binomial, . All perfect square trinomials with one variable have one root.

is a perfect square trinomial which would be transformed into the perfect square binomial of . Your root in this trinomial will be x=-7. The graph of this trinomial would look like this:

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

**Step 1: ** Transform your equation from standard form into a perfect square trinomial .

**Step 2: ** Transform the perfect square trinomial into a perfect square binomial, .

**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.

As you can see, it has one root and would be graphed like this:

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:

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

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

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.

**Step 4:** Use grouping to factor the expression.

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

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 into a perfect square binomial, .

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

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.

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

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

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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