7 min readConcept sheet1-mark MCQ; forming the equation is part of 3-mark and 5-mark word problems

Quadratic equations: standard form and forming them

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. A value of x that makes it true is a root.

Standard formax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0

Rootα is a root if aα² + bα + c = 0

Number of rootsAt most two

Roots and zeroesThe roots of ax² + bx + c = 0 are the zeroes of the polynomial ax² + bx + c

What makes an equation quadratic

An equation is quadratic if, after you simplify it, the highest power of x is 2. Write it in standard form ax² + bx + c = 0, where a ≠ 0. Always expand the brackets and bring all terms to one side first. Some equations look quadratic but are not, because the x² terms cancel. A root (or solution) is a value of x that makes the equation true.

Example 1: is it a quadratic equation?

Problem

Check whether each is a quadratic equation: (i) (x + 2)² = 3x − 1 (ii) (x − 1)(x + 3) = x² + 5 (iii) x(x + 1)² = x³ + 4

Working

(i) x² + 4x + 4 = 3x − 1, so x² + x + 5 = 0. The highest power is 2: quadratic.
(ii) x² + 2x − 3 = x² + 5, so 2x − 8 = 0. The x² terms cancel: not quadratic (it is linear).
(iii) x(x² + 2x + 1) = x³ + 4, so x³ + 2x² + x = x³ + 4, giving 2x² + x − 4 = 0. The x³ terms cancel: quadratic.

Answer

(i) and (iii) are quadratic equations; (ii) is not.

Example 2: checking roots

Problem

Check whether 3 and −1/2 are roots of 2x² − 5x − 3 = 0.

Working

x = 3: 2(9) − 5(3) − 3 = 18 − 15 − 3 = 0. So 3 is a root.
x = −1/2: 2(1/4) − 5(−1/2) − 3 = 1/2 + 5/2 − 3 = 3 − 3 = 0. So −1/2 is a root.

Answer

Both 3 and −1/2 are roots.

Forming an equation from a situation

1
Let the unknown be x. Say what it stands for, with units.
2
Write the other quantities in terms of x.
3
Write the condition in the question as an equation.
4
Expand and bring all terms to one side.
5★
Write it in the form ax² + bx + c = 0.

★ key step

Example 3: forming the equation for a garden

Problem

A rectangular vegetable garden in a Pune school is 3 m longer than it is wide. Its area is 108 m². Write a quadratic equation for its width.

Working

Let the width be x m. Then the length is (x + 3) m.
Area = length × breadth, so x(x + 3) = 108.
x² + 3x = 108
x² + 3x − 108 = 0
(Solving later gives x = 9, so the garden is 9 m by 12 m; 9 × 12 = 108.)

Answer

x² + 3x − 108 = 0, where x is the width in metres.

In real life

A farmer in Punjab may know the area of a rectangular field and how much longer it is than wide. Its width then satisfies a quadratic equation. The height of a ball thrown up in a cricket match also follows a quadratic.

Exam angle

1 marks: “Which of these is a quadratic equation: (x + 1)² = x² + 3, x² − 3x = (x − 1)², x(2x − 1) = 5 or x³ − x = 4?” x(2x − 1) = 5, which gives 2x² − x − 5 = 0. The first two become linear when the x² terms cancel, and the last is cubic.

2 marks: “The sum of a number and its reciprocal is 10/3. Write this as a quadratic equation.” x + 1/x = 10/3. Multiply by 3x: 3x² + 3 = 10x, so 3x² − 10x + 3 = 0.

1 marks: “If x = 2 is a root of x² + kx − 10 = 0, find k.” 4 + 2k − 10 = 0, so 2k = 6 and k = 3.

Remember it

Simplify first, then look for x²: if it survives, the equation is quadratic.