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 calculates 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

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

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.

solutions

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:

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

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
solution 1 by 0 solution 1 by 0 graph
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
solution 2 and 1 solution 2 and 1 graph
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
solution 1 and -1 solution 1 and -1 graph
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
solution -3 and 1

Below is a picture of the graph of the quadratic equation and its two solutions.

solution -3 and 1
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
solution -5 and -1
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
formula imaginary 1

Below is a picture of this quadratic's graph.

solutions imaginary 1
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