# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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References: Converse of a Theorem, Hypotenuse, Pythagorean Triples, Right Angle, Square, Square Root, Vertex.

PYTHAGOREAN TRIPLES

A right-angled triangle has three sides which obey the theorem of Pythagoras, c2 = a2 + b2. Any three whole numbers that fit this formula are called Pythagorean triples. For example, the numbers 3, 4, and 5 are Pythagorean triples, because 52 = 32 + 42. So are any multiples of 3,4, and 5, like 6, 8, and 10, or 30,40, and 50.

PYTHAGOREAN TRIPLES 357

There is an infinite number of Pythagorean triples, and some well-known ones are as follows:

♦ 5, 12, 13 (check: 132 = ? + 122)

♦ 7, 24, 25

♦ 8, 15, 17

♦ 9,40,41

♦ 11,60,61

Any multiples of these triples are also Pythagorean triples.

References: Pythagoras’ Theorem, Triangle.

Q

QUADRANTS

Reference: Circular Functions.

QUADRATIC EQUATIONS

A quadratic equation is an equation of the form

ax2 + bx + c = 0

where a is not equal to zero, and a, b, and c are real numbers. Examples of quadratic equations are

jc2 + 9x + 18 = 0 x2 + x — 6 = 0

x2 — 4x = —4 Which is the same as x2 — 4x + 4 = 0 2x2 — 6x + 4 = 0

x2 = 21 Which is the same as x2 + Ox — 21 = 0

x2 — 9 = 0 Which is the same as x2 + Ox — 9 = 0

4x2 = l(k Which is the same as 4x2 — 10x + 0 = 0

Some of the above equations may not appear to be of the form ax2 + bx + c = 0, but they can be rearranged into that form. To solve quadratic equations we first have to factorize the quadratic expression ax2 + bx + c. Before proceeding it may be necessary for you to study the entry Factorize. The theory behind solving quadratic equations is outlined here:

Suppose two terms A and B multiply together to give zero.

AB = 0

Then it follows that this equation is true if A = 0 or if B = 0.

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QUADRATIC EQUATIONS 359

If the two terms A and B are brackets, we can apply this theory in the following way:

Example 1. Solve (a) (x + 1)0 — 5) = 0, (b) 2x(x + 2) = 0.

Solution. For (a), write

0 + 1)0 - 5) = 0 0 + 1) = 0 or (x — 5) = 0 If AB = 0, then either A = 0 or 15 = 0

* + 1=0 or x — 5 = 0 We can dispense with the brackets

x = — 1 or x = 5 Solving linear equations

For (b), Write

2x(x + 2) = 0

2x = 0 or x + 2 = 0 If AB = 0, then either A = 0 or B = 0 x = 0 or x = — 2

In the next example the first stage is to express the quadratic equation in factorized form.

Example 2. Solve *2 + * - 6 = 0.

Solution. Write

x2 + x — 6 = 0 (x - 2)0 + 3) = 0

x — 2 = 0 or x + 3 = 0 x = 2 or x = — 3

Factorizing the quadratic

Example 3. Solve x2 -4x = -4.

Solution. Write

x2 -4x = -4

x2 - 4x + 4 = 0

(x - 2)0 - 2) = 0

Adding 4 to both sides of equation to make it = 0 Factorizing the quadratic

x — 2 (twice)

360 QUADRATIC EQUATIONS

When the two brackets are the same there will be only one solution, or we could say there are two equal solutions.

Example 4. Solve 2x2 - 6x + 4 = 0.

Solution. Write

2x2 — 6x + 4 = 0

2(x2 — 3x + 2) = 0 2 is a common factor; always look for these first

2(x — l)(x — 2) = 0 Factorizing the quadratic

(x — l)(x — 2) = 0 Dividing both sides of the equation by 2

x — 1=0 or x—2 = 0 x = 1 or x = 2

Example 5. Solve x2 - 9 = 0.

Solution. Write

x — 3 = 0 or x+3=0 x = 3 or x = — 3

Example 6. Solve 4.x2 = lOx.

Solution. Write

4x2 = lOx

4x2 — lOx = 0 Subtracting lOx from both sides, to

x2 - 9 = 0

(x - 3)(x + 3) = 0

Factorizing by difference of two squares

make equation = 0

2x(2x — 5) = 0

2x = 0 or 2x — 5 = 0

2x is a common factor

x = 0 or x = 2|

Solving the linear equations

Summary:

Always make sure the quadratic equation = 0. Look for common factors first.

QUADRATIC FACTORS 361

♦ Answers may be left as fractions or decimals, but if decimals are used, some may need rounding off.

♦ There are three kinds of factors to know: common, quadratic, and difference of two squares.

There are three types of quadratic equations that are not solved by factorizing, and their solutions are explained here.

Example 7. Solve jc2 = 21.

Solution. Write

JC2 = 21

= ±V2l Reduce x2 to x by taking the square root of both

sides of the equation

x = ±4.58 (to 2 dp) When we take the square root we get ±

Example 8. Solve = 9.

Solution. Write

*Jx = 9

(v^)2 = 92 Make sfx into a: by squaring both sides of the equation

jc = 81

Example 9. Solve (x - 3)2 = 16.

Solution. Write (x - 3 )2 = 16

sj(x — 3)2 = ±VT6 Taking the square root of both sides of the

equation

jc - 3 = ±4

x — 3 = +4 or x — 3 = —4 Separating out into two equations

x =1 or x = — 1

To solve other quadratic equations that do not factorize, see the entry Quadratic Formula for the method.

QUADRATIC FACTORS

References: Factor, Quadratic Equations.

362 QUADRATIC GRAPHS

QUADRATIC FORMULA

The general form of a quadratic equation is

ax2 + bx + c = 0

The coefficient of a:2 is a, the coefficient of x is b, and the constant term is c. The two roots of this quadratic equation are given by the formula

—b ± \J b2 — 4ac

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