Quadratic Equations - Formula, Solving, Examples

A quadratic equation is an equation of the form (ax^2 + bx + c = 0) where (a \neq 0). This article covers the standard form, the three methods for solving (factoring, completing the square, and the quadratic formula).

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation whose highest power of the variable is 2. The standard form is:

[ax^2 + bx + c = 0, \quad a \neq 0]

The two solutions (called roots or zeros) are given by the quadratic formula:

[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}]

A quadratic equation has up to two solutions. The graph of the corresponding function (f(x) = ax^2 + bx + c) is a parabola — a U-shaped curve that opens upward when (a > 0) and downward when (a < 0).

The Three Methods for Solving Quadratic Equations

Three methods solve every quadratic equation. Each one works for different situations.

Method 1: Factoring

Factoring works when the quadratic can be split into two linear factors with integer coefficients. Solve (x^2 - 5x + 6 = 0).

1. Find two numbers that multiply to (c = 6) and add to (b = -5). They are -2 and -3.
2. Rewrite the quadratic as a product: [x^2 - 5x + 6 = (x - 2)(x - 3) = 0]
3. A product equals zero only when at least one factor is zero. So (x - 2 = 0) or (x - 3 = 0), giving: [x = 2 \quad \text{or} \quad x = 3]

Method 2: Completing the Square

Completing the square is al-Khwarizmi's original method that works for every quadratic. Solve (x^2 + 6x - 7 = 0).

1. Move the constant to the RHS: [x^2 + 6x = 7]
2. Take half the coefficient of x — half of 6 is 3 — and square it: 9. Add this to both sides: [x^2 + 6x + 9 = 7 + 9]
3. The LHS is now a perfect square trinomial: [(x + 3)^2 = 16]
4. Take square roots of both sides: [x + 3 = \pm4]
5. Solve each case: [x = -3 + 4 = 1 \quad \text{or} \quad x = -3 - 4 = -7]

Method 3: The Quadratic Formula

The quadratic formula is the most general method — derived from completing the square on the general form (ax^2 + bx + c = 0).

Solve (2x^2 - 7x + 3 = 0). Identify (a = 2), (b = -7), (c = 3).

1. Compute the discriminant: [b^2 - 4ac = (-7)^2 - 4(2)(3) = 49 - 24 = 25]
2. Substitute into the formula: [x = \frac{-(-7) \pm \sqrt{25}}{2(2)} = \frac{7 \pm 5}{4}]
3. Compute both solutions: [x = 3 \quad \text{or} \quad x = \frac{1}{2}]

The Discriminant — Why It Matters

The discriminant, written (\Delta), tells you how many solutions the quadratic equation has:

What Are the Sum and Product of Roots? (Vieta's Formulas)

For a quadratic equation with roots (\alpha) and (\beta), two relationships hold:

[\alpha + \beta = -\frac{b}{a} \qquad \alpha \beta = \frac{c}{a}]

Where Quadratics Show Up in the Real World

• Projectile motion. A ball thrown into the air follows the path (y = -\frac{1}{2} g t^2 + v_0 t + h_0) — a quadratic in t.
• Area problems. A rectangle with length (x + 5) and width (x) has area (x(x + 5) = x^2 + 5x).
• Optimisation. The maximum or minimum value of a quadratic occurs at the vertex, (x = -\frac{b}{2a}).

Common Mistakes with Quadratic Equations

1. Forgetting the ± in the quadratic formula.
2. Sign error with -b when b is already negative.
3. Dividing both sides by a variable, losing solutions.

The Mathematicians Who Shaped Quadratic Equations

• Brahmagupta — First to give a systematic rule for solving quadratic equations.
• Muhammad ibn Musa al-Khwarizmi — Developed the geometric method of completing the square.
• Bhaskara II — Gave the modern quadratic formula and explained the role of the discriminant.

Frequently Asked Questions

What is a quadratic equation in simple terms? A quadratic equation is where the highest power of the variable is 2. It has up to two solutions.

What is the quadratic formula? The quadratic formula solves any quadratic equation in standard form, given by (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

What are the three methods to solve a quadratic equation? Factoring, completing the square, and the quadratic formula.