﻿ 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.

$$y = 5x^2 + 2x + 5 \\ y = 11x^2 + 22 \\ y = x^2 - 4x +5 \\ y = -x^2 + + 5$$

#### Non Examples

$$y = 11x + 22 \\ y = x^3 -x^2 +5x +5 \\ y = 2x^3 -4x^2 \\ y = -x^4 + 5$$

#### 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

Use the formula to solve theQuadratic Equation: $$y = x^2 + 2x + 1$$.

Just substitute a,b, and c into the general formula:

$$a = 1 \\ b = 2 \\ c = 1$$

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

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

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

• a = 1
• b = − 1
• c = − 2
##### Practice 3

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

• a = 1
• b = 0
• c = −1
##### Practice 4

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.

##### Practice 5

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

• a = 1
• b = 4
• c = −5
##### Practice 6

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.