The Quadratic Formula

Cones, Cylinders, Prisms

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.

Use The Quadratic Formula Calculator

Use The Quadratic Formula Calculator to see Quadratic Formula in Action! Calculator will solve any quadratic equation you type in (even if roots are imaginary)

The solution of a quadratic equation is the value of x when you set the equation equal to zero
i.e. When you solve the following general equation: 0 = ax² + bx + c

Given a quadratic equation: ax ² + bx + c

The quadratic formula below will solve the equation for zero

The quadratic formula is : Picture of the quadratic formula

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 formuls is this song (pop goes the weasal)

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

toSolving Quadratic Equations