# Quadratic Equation

Let's begin by completing the square on a generic
quadratic equation .

**Example 0.1 **ax^{2} + bx + c = 0

This formula allows us to skip the algebra and go straight
to the solution.

That is for ax^{2} + bx + c = 0,

All we have to do is identify the values for a , b, and c
in the quadratic

equation and substitute them into the formula we derived.

**Example 0.2** Solve : x^{2} + 8x + 15 = 0

a = 1 b = 8 c = 15

Using

The solution set is { -5,-3}

**Example 0.3** Solve : x^{2} + 6x + 13 = 0

a = 1 b = 6 c = 13

Using

The solution set is { -3 ± 2i}

**Example 0.4** Solve : 3x^{2} = 4x - 6

so 3x^{2} - 4x + 6 = 0

a = 3 b = -4 c = 6

Using

The solution set is

In the previous examples we could have solved x ^{2} +8x+15 = 0 by factoring

over the real numbers . However, x^{2} + 6x + 13 = 0 could not be solved this

way. How do we know when we will get real number solutions to an equation

and when we will get complex numbers as solutions?

** Property 0.1** The** discriminant **of ax^{2} + bx + c = 0 is b^{2}
- 4ac. Notice

that this is the part of the quadratic equation
that appears

under the square root .

So when the discriminant is positive then the solutions will be real numbers.

When the discriminant is negative the solutions will be complex numbers.

Remember that when we graph equations , the output
corresponds to the

y-value of the graph. When we solve equations that are set equal to zero ,

we are really finding the x- intercepts . When solving quadratic equations,

ax^{2} + bx + c = 0, we always get 2 solutions. But these solutions manifest in

different ways .

• The blue graph crosses the x-axis twice. This means that
solving the

corresponding equation will give two distinct real numbers. For ex-

ample x^{2} - 8x + 15 = (x - 5)(x - 3) = 0. This happens when the

discriminant is greater than 0.

•
The green graph just touches the x-axis in one spot. This means the

graph has one real solution that is repeated. For example x^{2}-10x+25 =

(x - 5)(x - 5) = 0. This happens when the discriminant is equal to 0.

•
The red graph has no x-intercepts. It has 2 solutions but they are not

real numbers. For example x^{2} + 1 = (x + i)(x - i) = 0. This happens

when the discriminant is less than zero.

Given a solution set we can write a quadratic equation
that has the given

solution set.

**
Example 0.5** Write a quadratic equation with solution set { -2, 6}.

so x = -2 or x = 6

so x + 2 = 0 or x - 6 = 0

so (x + 2)(x - 6) = 0

so x

^{2}- 4x - 12 = 0

What is the solution set of 2(x + 2)(x - 6) = 2x

^{2}- 8x - 24 = 0?

What does this tell us?

For any real number a, a(x + 2)(x - 6) = 0 will have the same solution set.

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