What is a quadratic equation?
A quadratic equation in one variable x is any equation that can be written in the standard form
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
The condition a ≠ 0 is what makes it quadratic: the highest power of x is exactly 2. If a were 0, the x² term would vanish and you would be left with a linear equation.
A value of x that satisfies the equation is called a root (or solution). A quadratic equation has at most two real roots.
Some equations only reveal themselves as quadratics after simplifying. For example, (x + 2)² − 3 = x + 7 expands to x² + 4x + 4 − 3 = x + 7, which rearranges to x² + 3x − 6 = 0. On the other hand, x(x + 1) = x² + 5 simplifies to x − 5 = 0, which is linear, because the x² terms cancel.
Method 1: Factorisation
If you can split the quadratic into two linear factors, use the zero-product rule: if p × q = 0, then p = 0 or q = 0.
To factorise ax² + bx + c, look for two numbers whose product is a × c and whose sum is b, then split the middle term.
Example. Solve 2x² − 5x + 3 = 0.
a × c = 2 × 3 = 6. We need two numbers with product 6 and sum −5: these are −2 and −3.
Split the middle term: 2x² − 2x − 3x + 3 = 0.
Group: 2x(x − 1) − 3(x − 1) = 0, so (2x − 3)(x − 1) = 0.
Therefore x = 3/2 or x = 1.
Always check by substituting back: for x = 1, 2(1) − 5(1) + 3 = 0. ✓
Method 2: Completing the square
This method works for every quadratic, and it is how the quadratic formula is derived.
Example. Solve x² + 6x − 7 = 0.
Move the constant: x² + 6x = 7.
Add the square of half the coefficient of x, that is (6/2)² = 9, to both sides: x² + 6x + 9 = 16.
The left side is now a perfect square: (x + 3)² = 16.
Take square roots: x + 3 = ±4, so x = 1 or x = −7.
If the coefficient of x² is not 1, divide the whole equation by a first.
Method 3: The quadratic formula
Completing the square on ax² + bx + c = 0 in general gives the quadratic formula:
x = [−b ± √(b² − 4ac)] / 2a
Example. Solve 3x² − 4x − 2 = 0.
Here a = 3, b = −4, c = −2, so b² − 4ac = 16 + 24 = 40.
x = [4 ± √40] / 6 = [4 ± 2√10] / 6 = (2 ± √10) / 3.
These roots are irrational, which is why factorisation would not have found them easily.
The discriminant and the nature of roots
The expression under the square root, D = b² − 4ac, is called the discriminant. Its sign tells you what kind of roots to expect, without solving:
Value of D | Nature of roots |
|---|---|
D > 0 | Two distinct real roots |
D = 0 | Two equal real roots, each equal to −b/2a |
D < 0 | No real roots |
Example. For what value of k does x² − kx + 9 = 0 have equal roots?
Equal roots need D = 0: k² − 4(1)(9) = 0, so k² = 36 and k = ±6.
Word problems: setting up the equation
Many board questions describe a situation in words. The skill is translating it into an equation.
Example. The product of two consecutive positive integers is 306. Find them.
Let the integers be x and x + 1. Then x(x + 1) = 306, so x² + x − 306 = 0. We need two numbers with product −306 and sum 1: these are 18 and −17. So (x + 18)(x − 17) = 0, giving x = 17 (we reject −18 because the integers are positive). The integers are 17 and 18.
Always check whether a root makes sense in the context. Lengths, ages and counts cannot be negative.
Common mistakes to avoid
Dividing both sides by x. This loses the root x = 0. Factor out x instead.
Forgetting the ± sign when taking a square root.
Using the formula before writing the equation in standard form (everything on one side, equal to 0).
Getting the sign of b wrong: in x² − 5x + 6 = 0, b is −5, not 5.
Quick summary
Standard form: ax² + bx + c = 0 with a ≠ 0.
Try factorisation first; use the formula when factors are not obvious.
D = b² − 4ac decides the nature of roots: positive, zero or negative.
In word problems, reject roots that do not fit the situation.