What makes an equation quadratic

A charity wants a prayer hall with a floor of 300 m², its length 1 m more than twice its breadth. Call the breadth x. The length is 2x + 1, so x(2x + 1) = 300, which is 2x² + x – 300 = 0. An x² has appeared: this is a quadratic equation.

The standard form: ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. Why a ≠ 0? If a were 0, the x² term would vanish and the equation would be linear.

Simplify first, then decide. Some equations are not what they look like:

• x(x + 1) + 8 = (x + 2)(x – 2) looks quadratic. Open the brackets: x² + x + 8 = x² – 4. The x² cancels, leaving x + 12 = 0, so it is not quadratic.
• (x + 2)³ = x³ – 4 looks cubic. Open it: x³ + 6x² + 12x + 8 = x³ – 4. The x³ cancels, leaving 6x² + 12x + 12 = 0, so it is quadratic.

Turning words into an equation. Let x be what is asked, then write each sentence in symbols:

• John and Jivanti have 45 marbles together. If John has x, Jivanti has 45 – x. Each loses 5, and the product of what's left is 124: (x – 5)(40 – x) = 124, which gives x² – 45x + 324 = 0.
• Two consecutive positive integers: x and x + 1. Two consecutive odd numbers: x and x + 2.

A bit of history: solving quadratic equations in general is often credited to Indian mathematicians. Brahmagupta (CE 598–665) gave an explicit formula for ax² + bx = c, and Sridharacharya (CE 1025) derived the formula we use today.