The Number That Predicts the Answer Quadratic Equations

A rectangular prayer hall needs to have its length twice its breadth plus one metre, and cover an area of exactly 300 square metres. Writing the breadth as x metres makes the length 2x + 1, and the area condition becomes x(2x + 1) = 300, which expands to 2x^2 + x - 300 = 0. A squared term has appeared where a straightforward equation used to be enough, and this squared-term equation is called a quadratic equation: any equation that can be rearranged into the standard form ax^2 + bx + c = 0, with a not equal to 0 (otherwise the x^2 term vanishes and it collapses back to a linear equation).

Checking whether an equation genuinely counts as quadratic sometimes needs real care. Take x(x+1) + 8 = (x+2)(x-2): expanding the left side gives x^2 + x + 8, and expanding the right side gives x^2 - 4. Setting them equal, x^2 + x + 8 = x^2 - 4, the x^2 terms on both sides cancel out completely, leaving just x + 12 = 0 -- a linear equation in disguise, not a quadratic one at all, despite every piece of the original equation looking squared. Genuinely confirming an equation is quadratic means simplifying it all the way down first, not just glancing at its ingredients.

Quadratic equations have a genuinely old history. The Babylonians could already solve specific numerical quadratics using geometric methods, and Euclid developed his own geometric approach later. In India, Brahmagupta (598-665 CE) gave an explicit formula for solving ax^2 + bx = c, and Sridharacharya (around 1025 CE) derived what is now called the quadratic formula by completing the square, a derivation later quoted directly by Bhaskara II. Around the same era, Al-Khwarizmi (whose name gave rise to the word 'algorithm') worked through similar methods in the Arabic mathematical tradition, and Abraham bar Hiyya Ha-Nasi's Liber Embadorum (1145 CE) carried a complete solution into European mathematics.

A root of a quadratic equation is precisely a value of x that makes the equation true -- which is exactly the same idea as a zero of the corresponding polynomial, the concept from the previous chapter. Splitting the middle term is the standard route: for 2x^2 - 5x + 3 = 0, two numbers are needed that multiply to 2 x 3 = 6 and add to -5, namely -2 and -3. Rewriting -5x as -2x - 3x gives 2x^2 - 2x - 3x + 3 = 0, which factors to 2x(x-1) - 3(x-1) = 0, so (2x-3)(x-1) = 0. Either factor being zero solves the equation, giving x = 3/2 or x = 1. Solving the prayer-hall equation, 2x^2 + x - 300 = 0, the same way gives a breadth of 12 metres and a length of 25 metres.

Factorisation works cleanly, but not every quadratic factors neatly with whole numbers, so a general method is needed. Starting from ax^2 + bx + c = 0, dividing through by a and completing the square on the x-terms eventually isolates x as x = (-b +/- sqrt(b^2 - 4ac)) / 2a -- the quadratic formula, working for every quadratic equation with real coefficients, whether or not it factors nicely. The expression under the square root, b^2 - 4ac, is called the discriminant, and its own sign, before any square root is even taken, already reveals what kind of roots are coming.

The discriminant D = b^2 - 4ac splits into exactly three cases. If D is positive, its square root is a genuine real number, giving two distinct real roots. If D is exactly zero, the plus-or-minus in the formula collapses to a single value, giving one repeated real root (both roots coincide). If D is negative, its square root isn't a real number at all, so the equation has no real roots whatsoever. A circular park of diameter 13 metres, with two gates whose distances from a pole differ by 7 metres, leads to the equation x^2 + 7x - 60 = 0, whose discriminant works out to 289 -- positive, confirming two distinct real solutions in advance, which turn out to be 5 metres and 12 metres (forming a 5-12-13 right triangle with the park's own diameter, via the Baudhayana-Pythagoras theorem).