Solving Quadratic Equations: Three Methods
A quadratic has exactly two solutions (real or complex). There are three standard methods — and knowing which to reach for first will save you significant time.
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form:
where a, b, and c are real numbers and a ≠ 0. The exponent 2 on x is what makes it quadratic (from the Latin quadratus, "square"). The solutions — the values of x that make the equation true — are called roots.
Method 1 — Factoring
Factoring is the fastest method when it works. You rewrite the left side as a product of two binomials, then use the zero-product property: if A × B = 0, then A = 0 or B = 0.
Example: Solve x² + 5x + 6 = 0.
Find two numbers that multiply to 6 and add to 5. Those are 2 and 3.
Set each factor to zero: x + 2 = 0 gives x = −2; x + 3 = 0 gives x = −3.
When to use it: When the coefficients are small integers and you can spot the factor pair quickly. If no integer pair works after about 30 seconds of searching, move to a different method.
Method 2 — Completing the Square
Completing the square transforms the equation into a perfect square trinomial, which can then be solved with a square root. It always works and also reveals the vertex form of the parabola — useful in many applications.
Example: Solve x² + 6x + 5 = 0.
- Move the constant to the right: x² + 6x = −5
- Take half the coefficient of x, square it: (6/2)² = 9. Add to both sides:
x² + 6x + 9 = −5 + 9 = 4 - Factor the left side as a perfect square: (x + 3)² = 4
- Take square roots: x + 3 = ±2
- Solve: x = −3 + 2 = −1, or x = −3 − 2 = −5
When to use it: When a = 1 and you prefer an algebraic derivation, or when the problem asks for vertex form. Also the method to use when deriving the quadratic formula itself.
The number you add to both sides is always (b/2a)². For standard form with a = 1, that simplifies to (b/2)². This single step is the whole trick — after that, everything is square roots and arithmetic.
Method 3 — The Quadratic Formula
The quadratic formula works for every quadratic, no matter the coefficients. It is derived by completing the square on the general form ax² + bx + c = 0:
Example: Solve 2x² − 3x − 2 = 0.
Here a = 2, b = −3, c = −2.
x = (3 + 5)/4 = 2, or x = (3 − 5)/4 = −1/2.
The Discriminant: Knowing What Kind of Roots to Expect
The expression under the square root — b² − 4ac — is called the discriminant. It tells you about the roots before you finish the calculation:
- Discriminant > 0: two distinct real roots
- Discriminant = 0: exactly one real root (a repeated root)
- Discriminant < 0: two complex (non-real) roots
For example, x² + x + 1 = 0 has discriminant 1 − 4 = −3. Since −3 < 0, there are no real solutions — the parabola never crosses the x-axis.
Which Method Should You Choose?
- Check for easy factoring first (5–10 seconds). If you see integer pairs immediately, factor.
- If a = 1 and integers don't cooperate, complete the square.
- When a ≠ 1 or when coefficients are messy decimals or fractions, go straight to the quadratic formula.
- Always verify by substituting your roots back into the original equation.
Summary
Quadratic equations have three standard solution methods. Factoring is fastest when it applies. Completing the square is systematic and illuminates the structure of the parabola. The quadratic formula is the universal fallback — memorize it and you can always find the roots. The discriminant (b² − 4ac) previews whether roots are real or complex before you finish calculating. Mastering all three methods gives you flexibility and a deeper understanding of why the algebra works.