Quadratic Equations: Solving Methods with Examples

A quadratic equation is an equation of the form ax squared + bx + c = 0, where a is not zero. Because the highest power is 2, a quadratic equation can have up to two solutions, called roots. These roots are the x-values where the parabola y = ax squared + bx + c crosses the x-axis.

This matters because quadratics model an enormous range of phenomena: projectile motion, areas, profits, and signal curves. Solving the equation answers concrete questions like when a ball lands or at what price profit is zero.

Key Concepts

• Standard form: ax squared + bx + c = 0. Always rearrange into this form before choosing a method.
• Roots (solutions, zeros): the x-values satisfying the equation; a quadratic has at most two.
• Discriminant: D = b squared - 4ac, which reveals the nature of the roots before you solve.
• Solution methods: factorisation (fastest when it works), completing the square, the quadratic formula (always works), and graphing.

Formulas

• Quadratic formula: x = (-b plus or minus root of (b squared - 4ac)) / (2a). Valid whenever a is not zero.
• Discriminant: D = b squared - 4ac. If D is positive there are two distinct real roots; if zero, one repeated root; if negative, no real roots.
• Sum and product of roots: for roots r1 and r2, r1 + r2 = -b/a and r1 times r2 = c/a. Handy for quick checks.

Worked Example

Solve 2x squared - 7x + 3 = 0.

1. Identify a = 2, b = -7, c = 3.
2. Discriminant: D = (-7) squared - 4(2)(3) = 49 - 24 = 25.
3. Root of D = 5, so x = (7 plus or minus 5) / 4.
4. x = 12/4 = 3 or x = 2/4 = 1/2.

Check x = 3: 2(9) - 7(3) + 3 = 18 - 21 + 3 = 0. Check x = 1/2: 2(1/4) - 7/2 + 3 = 1/2 - 7/2 + 3 = -3 + 3 = 0. Both roots verified.

Common Misconceptions

• Dividing both sides by x is a safe first step. Wrong. If x = 0 is a root, dividing by x throws it away. Factor out x instead of dividing by it.
• Every quadratic has two real solutions. Wrong. When the discriminant is negative there are no real roots, as with x squared + x + 1 = 0.
• The formula works on any arrangement. Only after rearranging into ax squared + bx + c = 0, so that a, b and c (with their signs) are read correctly.

Real-World Applications

• In physics, the equation of projectile motion is quadratic in time; its roots give the launch and landing moments.
• In business, profit as a function of price is often quadratic, and its roots are the break-even prices.
• In engineering, resonance and optimisation problems frequently reduce to solving a quadratic equation.

To recap: write the equation in standard form, try factorisation first, and fall back on the quadratic formula which never fails. Use the discriminant to predict the roots and the sum-product relations to check them.