# 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:

| Discriminant | Number of real solutions | What the graph looks like |
| --- | --- | --- |
| \(\Delta > 0\) | Two distinct real solutions | The parabola crosses the x-axis at two points |
| \(\Delta = 0\) | One repeated real solution | The parabola touches the x-axis at one point (vertex) |
| \(\Delta < 0\) | No real solutions (two complex) | The parabola doesn't touch the x-axis |

## 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.