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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 ax2 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+ 10x to make it a perfect square?

Solution

  • Let the number to be added be a constant c.
  • Then x2+ 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 x2 + _ + 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– 12x + 9 = (2x – 6)2
9x– 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.

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

Example

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

Solution

  • Let the term to be added be bx.
  • Then, 25x+ 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?

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

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 16x2

Solutions of Quadratic Equations by Completing the Square Method

Example

Solve x+ 5x+ 1 = 0 by completing the square.

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

solution

x+ 5x+ 1 = 0 Write original equation.

x+ 5x = -1 Write the left side in the form xbx.

x2+ 1 0+ (5/2)2= (5/2)2 – 1 Add (5/2)to both sides

x2+ 1025/4 21/4

>

(x + 5/221/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

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

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

Example

>

Solve 2x2 + 4x+ 1 = 0 by completing the square

Solution

2x2 + 4x =-1 make coeffiecient of xone by dividing both sides by 2

>

x+ 2x = –1/2

x+ 2x + 1 = –1/2 + 1

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

Adding 1 to complete the square on the LHS
( x + 1)= 1
2
x + 1 = ±√1/2
x = -1 ± √0.5
= -1 ± 0.7071
x = 0.2929 or – 1.7071

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

FORM 1 END TERM EXAMINATIONS FREE DOWNLOAD

FORM 2 END TERM EXAMINATIONS FREE DOWNLOAD

FORM 3 END TERM EXAMINATIONS FREE DOWNLOAD

FORM 4 END TERM EXAMINATIONS FREE DOWNLOAD

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