Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula

Standard Form of a Quadratic Equation

A quadratic equation is any equation that can be written in the form:

Here a, b, c are constants and a ≠ 0 (otherwise the equation is linear, not quadratic). Before applying any solution method, rearrange the equation so one side equals zero — this sets up the zero-product property for factoring and matches the setup of the quadratic formula.

Three solution methods cover every quadratic: factoring, completing the square, and the quadratic formula. The discriminant tells you what kind of solutions exist before you compute them.

Solving by Factoring

When the quadratic factors nicely, factoring is the fastest method. Set the equation to zero, factor, then apply the zero-product property: a product equals zero when at least one factor is zero.

Example: Solve x^2 + 5x + 6 = 0.

See the factoring polynomials guide for all factoring techniques. If the trinomial doesn't factor over the integers, use completing the square or the quadratic formula.

Completing the Square

Completing the square rewrites x² + bx + c as (x + b/2)² + k. This is the technique behind the quadratic formula and the foundation for graphing parabolas in vertex form.

Steps:

1. Move the constant to the right side.
2. If a ≠ 1, divide everything by a.
3. Take half of the x-coefficient, square it, and add it to both sides.
4. Factor the left side as a perfect square and take square roots.

Example: Solve x^2 + 6x - 7 = 0.

Move the constant:

Half of 6 is 3, squared is 9. Add 9 to both sides:

Take square roots:

The Quadratic Formula and Its Derivation

The quadratic formula gives the solutions to any quadratic equation in standard form:

Derivation sketch: apply completing the square to the general form ax² + bx + c = 0. Divide by a, move c/a to the right, complete the square with (b/2a)², and solve. The result is the formula above.

Example: Solve 2x^2 - 3x - 5 = 0.

Substitute a = 2, b = -3, c = -5:

The quadratic formula works for every quadratic — use it whenever factoring fails or when you need exact decimal answers.

The Discriminant: How Many Solutions?

The discriminant is the expression under the square root in the quadratic formula: \Delta = b^2 - 4ac. Its sign tells you how many real solutions the equation has before you compute them.

• Δ > 0: two distinct real solutions (parabola crosses x-axis twice)
• Δ = 0: one repeated real solution (parabola touches x-axis at its vertex)
• Δ < 0: two complex conjugate solutions (parabola never crosses x-axis)

If the discriminant is a perfect square, the quadratic factors nicely over the integers — this is a quick way to decide whether factoring will work.

Complex Roots

When the discriminant is negative, the quadratic has no real solutions — but it always has two complex solutions (conjugates of each other). Complex roots use the imaginary unit i, where i² = -1.

Example: Solve x^2 + 2x + 5 = 0.

Discriminant: 4 - 20 = -16. Apply the quadratic formula:

The two solutions -1 + 2i and -1 - 2i are complex conjugates — real quadratics always produce complex roots in conjugate pairs.

Applications and Word Problems

Quadratic equations model many real situations: projectile motion, area problems, optimization, and revenue/profit calculations.

Projectile example: A ball is thrown upward from an 80-foot building with an initial velocity of 64 ft/s. Its height is:

When does it hit the ground? Set h(t) = 0 and solve:

(The other algebraic solution t = -1 is rejected because time cannot be negative.) Quadratic word problems often yield two algebraic solutions — always check which one makes sense in the problem's context.

Common Mistakes

Practice Problems

Solve each equation using the best method (factoring, completing the square, or the quadratic formula).

1. x^2 - 7x + 12 = 0
2. 2x^2 + 5x - 3 = 0
3. x^2 + 4x - 1 = 0
4. x^2 + 6x + 10 = 0
5. 3x^2 = 2x + 1
6. x^2 - 2x - 15 = 0

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