﻿ 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 does this formula tell us?

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.