A square room has a floor area of 16 m². How long is each wall? If you call the length of a wall x, then x² = 16. Which number gives 16 when you square it? The number 4, but also −4, because (−4)² = 16. For a wall only 4 m makes sense, but the equation itself has two solutions.
An equation where the unknown is squared is a quadratic equation. In this article you will solve the simplest ones, the incomplete ones. You don't need any formula for them.
Every quadratic equation can be rearranged into the form
ax² + bx + c = 0, where a ≠ 0.
The numbers a, b, c are the coefficients:
a is the number in front of x² (the x² term),
b is the number in front of x (the x term),
c is the number without x (the constant term).
For example, in the equation 2x² − 3x + 1 = 0 you have a = 2, b = −3, c = 1.
The equation is incomplete when b = 0 or c = 0, so when the x term or the constant term is missing. The number a must not be missing, otherwise the equation would not be quadratic.
Equations with no x term
An equation of the form ax² + c = 0 (b = 0) is solved like this:
Get x² on its own on one side.
Take the square root. Don't forget there are two solutions: a positive one and a negative one. You write this with the sign ± ("plus or minus").
Worked example 1
x² − 16 = 0
x² = 16
x = ±4, so x₁ = 4, x₂ = −4.
Check: 4² − 16 = 0 ✓, (−4)² − 16 = 16 − 16 = 0 ✓.
Worked example 2
3x² − 75 = 0
3x² = 75
x² = 25 (we divided both sides by three)
x₁ = 5, x₂ = −5
Worked example 3
4x² − 9 = 0
4x² = 9
x² = 9/4
x = ±3/2, so x₁ = 3/2, x₂ = −3/2.
Worked example 4
x² + 4 = 0
x² = −4
The square of a real number is never negative: a positive number squared is positive, a negative number squared is positive too, and zero gives zero.
The equation has no real solutions.
💡 If you get x² = 7, the solutions are x = ±√7. The square root of a number that is not a square number is still a solution. You just can't write it as a whole number.
Equations with no constant term
An equation of the form ax² + bx = 0 (c = 0) has x in both terms. Take x out as a common factor:
x · (ax + b) = 0
A product is zero exactly when at least one of its factors is zero. So either x = 0 or ax + b = 0.
Worked example 5
x² − 5x = 0
x · (x − 5) = 0
x = 0 or x − 5 = 0
x₁ = 0, x₂ = 5
Check: 5² − 5 · 5 = 25 − 25 = 0 ✓.
Worked example 6
2x² + 3x = 0
x · (2x + 3) = 0
x = 0 or 2x + 3 = 0, so 2x = −3
x₁ = 0, x₂ = −3/2
⚠️ Never divide the equation x² − 5x = 0 by x! You would get x − 5 = 0 and lose the solution x = 0. You can't divide by zero, and x can be zero.
Equations you need to rearrange first
Sometimes an equation doesn't look incomplete, but it becomes incomplete once you rearrange it.
Worked example 7
(x − 3)² = 16
The expression in the bracket gives 16 when squared, so it equals 4 or −4:
