Solving Quadratic Equations

 Factoring

 A quadratic equation is an equation that can be simplified to follow the pattern:

 y = a · x ² + b · x + c,

 where the letter x represents the input, the letter y represents the value of the output and the letters a, b and c are all numbers. Sometimes the numbers a, b and c are referred to as coefficients.

This way of writing the equation of a quadratic function is called the standard form of the quadratic function.

The Factored Form of a quadratic and what it can tell you about the graph

A second way to write the equation of a quadratic function is called the factored form. The factored form of a quadratic function follows the algebraic format:

y = a · (x - c) · (x - d).

As with the standard form, the sign (+ or -) of the number a tells you whether the quadratic function “smiles” (a > 0) or “frowns” (a < 0). Sometimes you will see an example of a factored form that does not appear to have a value of a, such as:

 y = (x -1)  ·  (x + 2).

In this case, the value of a is equal to one (which is positive) and the graph of the quadratic will smile.

The factored form of a quadratic equation also tells you where the x-intercepts (sometimes called the roots or zeros of the quadratic function are located). The xintercepts of the equation are the x-values that will make y = 0.

For example, the x-intercepts of the quadratic:

 y = (x -1)  ·  (x + 2)

are x = 1 and x = -2 as plugging either of these two values into the quadratic equation will make y equal to zero.

Example

Figure 1 (below) shows the graph of a quadratic function. Find the equation of this quadratic and express your answer in both factored and standard forms.

Figure 1: Find the formula of this quadratic function.

Solution

Figure 1 clearly shows both the of the x-intercepts of the quadratic function so it will be easiest to find the factored form of the quadratic first and then convert this to standard form by FOILing.

The x-intercepts of the quadratic shown in Figure 4 are located at x = 1 and x = 4. This means that the factored form of the quadratic function must look something like this:

 y = a · (x -1) · (x - 4).

The factored form must have a factor of (x - 1) to ensure that when you plug in x = 1 the value of y will be equal to zero. The factored form must also have a factor of (x - 4) to ensure that when you plug in x = 4 the value of y will be equal to zero.

To determine the numerical value of a you can plug in the x- and y-coordinates of any other point on the quadratic graph (i.e. any point other than one of the x-intercepts) and solve for a. Figure 4 shows that the point (0, -2) lies on the graph, so you can plug in x = 0 and y = -2 into the factored form. Doing this:

 -2 = a · (0 -1) · (0 - 4)

 -2 = a · 4

So, the equation of the quadratic function from Figure 4 (written in factored form) is:

To convert this equation to standard form, you can expand by FOILing and then simplify (if necessary). Doing this:

(Expand by FOILing)

(Multiply through by - beware of “-” signs)

Example

A quadratic equation is expressed in standard form as:

 y = 3 · x ² - 21 ·  x + 36.

Convert this formula to factored form.

Solution

When converting a quadratic function to factored form (or factoring), the first step is always to factor out by the number that is multiplying x ². In this case that number is 3. 

 y = 3 ·  (x ² - 7 ·  x +12).

 Next, you look at what is left inside the parentheses. What you are looking for is a pair of numbers that:

1. Add to give the coefficient of x that still remains inside the parentheses, and,

2. Multiply to give the constant number that still remains inside the parentheses.

In this particular example, we are looking for two numbers that will add to give -7 and that will multiply to give 12. Two numbers that fit this bill are -3 and -4, as:

-3 + -4 = -7

(-3)( -4) = 12.

The two numbers -3 and -4 are the numbers that are added to x in each of the factors of the factored form. As we have already worked out the value of the constant a (it is 3), the factored form for this quadratic will be:

y = 3 · (x + -3) · (x + -4) = 3 · (x - 3) · (x - 4).