The Quadratic Formula

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

What is the Quadratic Formula?

The quadratic formula is : Picture of the quadratic formula

What does this formula tell us?

The quadratic formula will calculate the solutions of any quadratic equation.

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. solutions
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:

example of quadratic formula

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

Picture of graph of  solved quadratic formula

Quadratic Formula Song

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

Practice Problems

Practice 1

Use the quadratic formula to find the solutions to the following equation: y = x² − 2x + 1 and its solution

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

Use the quadratic formula to find the solutions to the following equation: y = x² − x − 2 and its solution

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

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

Use the quadratic formula to find the solutions to the following equation: y = x² − 1 and its solution

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

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

Calculate the solutions of the the quadratic equation below by using the quadratic formula : y = x² + 2x − 3 and its solution

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

Use the quadratic formula to find the solutions to the following equation: y = x² + 4x − 5 and its solution

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

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

Use the quadratic formula to find the solutions to the following equation: y = x² − 4x + 5 and its solution

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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