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Quadratic Equations
Introduction
Quadratic Equations
Practice: Recognising Quadratic Equations
Solution of a Quadratic Equation by Factorisation
Practice: Solving by Factorisation
Nature of Roots
Practice: Nature of Roots
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Quadratic EquationsIntroduction

Reading time: ~15 min

A charity trust in your town is building a new prayer hall. They have already bought the carpet — one enormous roll that covers exactly 300 m² — and the carpet seller refuses to cut it. The architect adds one more rule: the hall's length must be one metre more than twice its breadth.

So the floor has to match the carpet exactly. How wide should the builders make the hall?

You are the architect. Fig. 4.1 below is the hall seen from above: the brown rectangle is the floor, with the walls drawn as its thick edges, and the red rectangle is the 300 m² carpet laid on it. Drag the breadth. When the floor is too small, the carpet spills past the walls; when the floor is too big, bare floor shows around the carpet. Find the breadth at which the carpet fits the floor exactly.

Breadth of the hall: ${w} m

You found it by trial and error: a breadth of 12 m. That worked because the answer happened to be a whole number. If the carpet were 310 m² instead, the perfect breadth would be about 12.2 m, and sliding in whole metres would never hit it. We need a method that reasons its way to the answer instead of guessing. That method is algebra.

Call the breadth of the hall x metres. Then its length is metres.

The floor area is breadth × length, so \text{area} = x(2x + 1) = square metres.

The carpet fits exactly when this area is 300, so 2x^2 + x = 300. Moving everything to one side gives .

In Chapter 2 you met polynomials, including the quadratic polynomial ax^2 + bx + c with a \neq 0. Set one of them equal to zero and you get a quadratic equation — exactly what our hall has produced. The name comes from quadratus, the Latin word for "square", because of that x^2 term. By the end of this chapter you will be able to solve this equation properly, and we will come back to the hall to do it.

A 4000-Year-Old Puzzle

Quadratic equations are ancient. On clay tablets from Babylon, written around 4000 years ago, scribes set puzzles like this one:

"Find two positive numbers whose sum is 10 and whose product is 21."

The smaller number is and the larger number is .

Without knowing it, you have just solved the quadratic equation x^2 - 10x + 21 = 0 — put in 3 or 7 and the left-hand side becomes zero. Every "given sum, given product" puzzle hides a quadratic of the form x^2 - px + q = 0. Remember this puzzle: later in the chapter it turns into the key for factorising.

Many people think the Babylonians were the first to solve quadratic equations. After them, the Greek mathematician Euclid found such unknown lengths with pure geometry — constructions that, in today's language, give the solutions of quadratic equations.

Solving the general quadratic is often credited to ancient Indian mathematicians. Brahmagupta (C.E. 598–665) wrote down an explicit rule for solving ax^2 + bx = c. Around C.E. 800 the Arab mathematician Al-Khwarizmi sorted quadratic equations into different types and studied each one. Then, around C.E. 1025, Sridharacharya found a formula for solving any quadratic by completing the square — we know it because Bhaskara II quoted it, and today we call it the quadratic formula. In C.E. 1145, Abraham bar Hiyya Ha-Nasi's book Liber embadorum, published in Europe, gave complete solutions of different quadratic equations.

Drag these milestones into the order in which they happened, earliest at the top.

Al-Khwarizmi sorts quadratic equations into types
Babylonian scribes set sum-and-product puzzles on clay tablets
Liber embadorum brings complete solutions to Europe
Euclid finds the unknown lengths using geometry
Sridharacharya derives the quadratic formula
Brahmagupta writes a rule for solving ax² + bx = c

Four thousand years, and the puzzle is still the same one you just solved. In this chapter you will learn what makes an equation quadratic, find its roots by factorising, and predict what kind of roots it has before solving it — with plenty of real-life puzzles along the way.

Sina