# Quadratic Expressions and Equations – Mathematics Form 3 Notes and K.C.S.E Topical Q&A

## Quadratic Expressions and Equations - Mathematics Form 3 Notes and K.C.S.E Topical Q&A

## Quadratic Expressions and Equations – Mathematics Form 3 Notes and K.C.S.E Topical Q&A

Quadratic Expressions and Equations – Mathematics Form 3 Notes and K.C.S.E Topical Q&A

**<<<Download>>>K.C.S.E TOPICAL QUESTIONS AND ANSWERS TESTED FROM THIS TOPIC**

Quadratic Expressions and Equations – Mathematics Form 3 Notes and K.C.S.E Topical Q&A

## Perfect Square

- Expressions which can be factorized into two equal factors are called perfect squares.

## Completing the Square

- Any quadratic expression can be simplified and written in the form ax
^{2}x bx + c where a, b and c are constant and a is not equal to zero. We use the expression (^{b}/_{2})^{2}= C to make a perfect square - We are first going to look for expression where coefficient of x = 1

**Example**

What must be added to x^{2 }+ 10x to make it a perfect square?

**Solution**

- Let the number to be added be a constant c.
- Then x
^{2}+ 10x + c is a perfect square. - Using (
^{b}/_{2})^{2} - (
^{10}/_{2})^{2}= c - c = 25 (25 must be added)

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**Example**

What must be added to x^{2} + _ + 36 to make it a perfect square

**Solution**

- Let the term to be added be bx where b is a constant
- Then x2+ bx +36 is a perfect square.
- Using ((
^{b}/_{2})^{2}= 36 ^{b}/_{2}=√36- b/
_{2}= ±6 b =1 2 x or -1 2 x

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We will now consider the situations where a≠ 1 and not equal to zero eg

4x^{2 }– 12x + 9 = (2x – 6)^{2}

9x^{2 }– 6x + 1 = (3x + 1)^{2}

In the above you will notice that (^{b}/_{2})^{2} = ac . We use this expression to make perfect squares where a is not one and its not zero.

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**Example**

What must be added to 25x^{2 }+ _ + 9 to make it a perfect square?

**Solution**

- Let the term to be added be bx.
- Then, 25x
^{2 }+ bx + 9 is a perfect square. - Therefore (
^{b}/_{2})^{2}= 25 x 9. - (
^{b}/_{2})^{2}= 225 ^{b}/_{2}= ±15- so b = 30 or – 30 The term to be added is thus30 or – 30.

**Example**

What must be added to _ – 40x + 25 to make it a perfect square?

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**Solution**

- Let the term to be added be ax2
- Then ax2– 40x + 25 is a perfect square.
- Using (
^{b}/_{2})^{2}= ac - (
^{-40}/_{2})^{2}= 25a - 400 = 25a
- a = 16 the term to be added is 16x
^{2}

### Solutions of Quadratic Equations by Completing the Square Method

**Example**

Solve x^{2 }+ 5x+ 1 = 0 by completing the square.

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**solution**

x^{2 }+ 5x+ 1 = 0 Write original equation.

x^{2 }+ 5x = -1 Write the left side in the form x^{2 }+ bx.

x^{2}+ 1 0x + (^{5}/_{2})^{2}= (^{5}/_{2})^{2} – 1 Add (^{5}/_{2})^{2 }to both sides

x^{2}+ 10x + ^{25}/_{4} = ^{21}/_{4}

(x + ^{5}/_{2}) = ^{21}/_{4 }Take square roots of each side and factorize the left side

x + ^{5}/_{2} =± √(^{21}/4)_{ }Solve for x.

= –^{5}/_{2} ± ^{4.583}/_{2} Simplify

= ^{0.417}/_{2} or ^{9.583}/_{2} Therefore x = – 0.2085 or 4.792

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The method of completing the square enables us to solve quadratic equations which cannot be solved by factorization.

**Example**

Solve 2x^{2} + 4x+ 1 = 0 by completing the square

**Solution**

2x^{2} + 4x =-1 make coeffiecient of x^{2 }one by dividing both sides by 2

x^{2 }+ 2x = –^{1}/_{2}

x^{2 }+ 2x + 1 = –^{1}/_{2} + 1

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Adding 1 to complete the square on the LHS

( x + 1)^{2 }= 1

2

x + 1 = ±√^{1}/_{2}

x = -1 ± √0.5

= -1 ± 0.7071

x = 0.2929 or – 1.7071