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Quadratic Equations

Example

A quadratic equation is expressed in standard form as:

y = 6 · x 2 +12 · x - 90.

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 = 6 · (x 2 + 2 · x -15).

Next, you look at what is left inside the parentheses. What you are looking for are two 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 +2 and that will multiply to give -15. Two numbers that fit this bill are -3 and +5, as:

-3 + 5 = 2

(-3)( 15) = -15.

The two numbers -3 and 5 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 6), the factored form for this quadratic will be:

y = 6 · (x + -3) · (x + 5) = 6 · (x - 3) · (x + 5) .

Some Common Factoring Patterns

x 2 - a 2 = (x - a) · (x + a) “Difference of two squares”

x 2 + 2· a · x + a 2 = (x + a) 2 “Perfect square I”

x 2 - 2 · a · x + a 2 = (x - a) 2 “Perfect square II”

Example

Convert the quadratic function:

y = 2· x 2 -16 · x + 42,

from standard to vertex form and locate the x- and y-coordinates of the vertex.

Solution

Once the formula for the quadratic function has been converted to vertex form:

y = a · (x - h) 2 + k,

we can find the vertex by checking the vertex form to find the values of h (which will be the x-coordinate of the vertex) and k (which will be the y-coordinate of the vertex).

Conversion of the formula from standard to vertex form is a four-step process called completing the square.

1. Factor out the coefficient of x 2 from all terms.

y = 2 · (x 2 - 8 · x + 21)

2. Add and subtract just the right amount* to create a perfect square

y = 2 · (x 2 - 8 · x +16 -16 + 21)

3. Factor the perfect square and combine the constants

y = 2· ((x - 4)2 -16 + 21)

y = 2 · ((x - 4)2 + 5)

4. Distribute the factor that is out in front of the equation

y = 2 · (x - 4)2 +10

The vertex form of the quadratic function is:

y = 2 · (x - 4)2 +10.

* To find just the right amount, you take the number that is left multiplying the x after Step 1 has been completed. Whatever this number is, divide the number by 2 and then take the square of what you are left with. This is just the right amount to create a perfect square.

 
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