﻿ The Quadratic Formula to solve quadratic equations Step by step with graphs to illustrate.

What it is, what it does, and how to use it

#### What is a quadratic equation?

A quadratic equation is an equation that can be written as ax ² + bx + c where a ≠ 0 In other words, a quadratic equation must have a squared term as its highest power Examples of quadratic equations

• y = 5x² + 2x + 5
• y = 11x² + 22
• y = x²− 4x +5
• y = −x² + 5
Examples of equations that are not quadratic
• y = 11x + 22
• y = x3 − 4x² +5x +5
• y = 2x3 − 4x²
• y = −x4 + 5

#### Ok, but what is a 'solution'?

Well a solution can be thought in two ways:

 Algebra: For any quadratic equation of the form f(x) = ax2+bx+c, the solution is when f(x) = 0. Geometry The solution is where the graph of a quadratic equation (a parabola) is intersects the x-axis. This, of course, only applies to real solutions.

### Example of the quadratic formula to solve an equation

Quadratic Equation: y = x² + 2x + 1, a = 1, b = 2, c = 1

Using the quadratic formula to solve this equation just substitute a,b, and c into the general formula:

Below is a picture representing the graph of y = x² + 2x + 1 and its solution

A catchy way to remember the quadratic formula is this song (pop goes the weasel)

### Practice Problems

In this quadratic equation, y = x² − 2x + 1 and its solution
• a =1
• b = − 2
• c = 1

In this quadratic equation,y = x² − x − 2 and its solution

• a =1
• b = − 1
• c = − 2

In this quadratic equation, y = x² − 1 and its solution

• a = 1
• b = 0
• c = −1

In this quadratic equation, y = x² + 2x − 3 and its solution

• a =1
• b = 2
• c = −3

Below is a picture of the graph of the quadratic equation and its two solutions In this quadratic equation, y = x² + 4x − 5 and its solution

• a =1
• b = 4
• c = −5

In this quadratic equation,y = x² − 4x + 5 and its solution

• a =1
• b = −4
• c = 5

Below is a picture of this quadratic's graph

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