The Quadratic Formula

It finds the values of x that solve any equation shaped like a x squared plus b x plus c equals zero.

x = -b ± √(b² - 4ac)2a

A quadratic draws a parabola. The formula finds where it crosses the x-axis: the two roots.

What each symbol means

a: the number in front of x² (it cannot be zero)
b: the number in front of x
c: the plain number with no x (the constant)
x: the unknown value you are solving for
±: plus-or-minus: you work it out twice, once with + and once with -

A quadratic equation is any equation you can write as ax² + bx + c = 0. Some quadratics are easy to solve by factoring, but many are not. The quadratic formula always works, every time, no guessing required.

Read off the three numbers a, b, and c from your equation, drop them into the formula, and it hands you the answers.

The plus-or-minus gives two answers: The ± symbol means a quadratic usually has two solutions: one using the plus, one using the minus. On the graph, those are the two spots where the curve crosses the x-axis.

A worked example

Solve x² + 5x + 6 = 0

Read off the numbers: a = 1, b = 5, c = 6.
Write the formula: x = -b ± √(b² - 4ac)2a.
Put in the numbers: x = -5 ± √(5² - 4(1)(6))2(1).
Work inside the root: 5² - 24 = 25 - 24 = 1, and √(1) = 1.
So x = -5 ± 12.
Plus: x = -5 + 12 = -42 = -2. Minus: x = -5 - 12 = -62 = -3.
The two solutions are x = -2 and x = -3.

The part under the root tells a story: The expression b² - 4ac is called the discriminant. If it is positive there are two answers, if it is zero there is one, and if it is negative there are no real answers (the parabola never touches the x-axis).

The quadratic formula solves ax² + bx + c = 0 every time.
It is x = -b ± √(b² - 4ac)2a.
The ± gives the two roots, where the parabola crosses the x-axis.
The discriminant b² - 4ac tells you how many real answers there are.

Where the formula comes from (completing the square)

Start with ax² + bx + c = 0 and divide everything by a: x² + bax + ca = 0.
Move the constant across: x² + bax = - ca.
Add ( b2a)² to both sides so the left side becomes a perfect square: (x + b2a)² = b² - 4ac4a².
Take the square root of both sides (this is where the ± appears): x + b2a = ± √(b² - 4ac)2a.
Subtract b2a and combine over one denominator: x = -b ± √(b² - 4ac)2a.

Practice questions

Question 1

Solve x² - 5x + 6 = 0.

Read off the numbers: a = 1, b = -5, c = 6.
Work out the discriminant b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1.
Put it into the formula: x = -(-5) ± √(1)2(1) = 5 ± 12.
Take the plus: x = 5 + 12 = 3. Take the minus: x = 5 - 12 = 2.
The two solutions are x = 2 or x = 3.

Answer: x = 2 or x = 3

Question 2

Solve x² - 4 = 0.

Read off the numbers: a = 1, b = 0, c = -4.
Work out the discriminant b² - 4ac = 0 - 4(1)(-4) = 16, and √(16) = 4.
Put it into the formula: x = 0 ± 42.
This gives x = 42 = 2 and x = -42 = -2.
The two solutions are x = 2 or x = -2.

Answer: x = 2 or x = -2

Question 3

Solve x² + 2x + 1 = 0.

Read off the numbers: a = 1, b = 2, c = 1.
Work out the discriminant b² - 4ac = 4 - 4(1)(1) = 0.
A discriminant of 0 means the ± adds nothing, so there is just one answer.
Put it into the formula: x = -2 ± 02 = -22 = -1.
So there is one repeated root, x = -1.

Answer: x = -1 (one repeated root)

Question 4

In 2x² + 3x - 2 = 0, what are a, b, and c?

Line the equation up with the standard shape ax² + bx + c = 0.
The number in front of x² is a, so a = 2.
The number in front of x is b, so b = 3.
The plain number on its own is c, so c = -2 (keep its minus sign).

Answer: a = 2, b = 3, c = -2

Question 5

What does a discriminant of -7 tell you?

The discriminant is the part under the square root, b² - 4ac.
Here it is -7, which is negative.
You cannot take the square root of a negative number and get a real result.
So the formula gives no real answers, and the parabola never crosses the x-axis.

Answer: There are no real solutions.