What is the Quadratic Formula?
The quadratic formula is:
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. 
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.
Quadratic Formula Song
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:
![solution 1 by 0]()
![solution 1 by 0 graph]()
- 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.
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