A quadratic equation is an expression of the form ax^{2} + bx + c = 0 in which a, b & c are numerals; and also the highest power of x is 2 & that the power of x will neither be fractions nor negatives. Quadratic equations can be solved using the method of factorization, completing the square, quadratic formula& graphical method

Steps in solving quadratic equation: (1)examine the middle term whose power of x is 1. (2) Find the product of the first & last term. (3) Find two terms whose sum is equal to the middle term & product is equal to the value of the product of the first & last term (4) Replace the middle term by two the two terms in step 3. (5) Factorize the first two & last two terms (6) equate the linear factors to zero to find the value of x.

Example – Solve by factorization: X^{2} + 7X + 10 = 0

Solution

X^{2} + 7X + 10 = 0

X^{2} +2X + 5X + 10 = 0

X(X + 2) + 5(X + 2) = 0

(X+2) (X + 5) = 0

X + 2 = 0

X = -2 OR

X + 5 = 0

X = -5

Hence X = -2 or -5

## Factorization of Quadratic Expressions.

Factorize the following:

- 6a
^{2}+ 15a + 9 - 6a
^{2}– 19ax – 36x^{2} - (5x – 1) ( x – 3) – ( x – 5) ( x – 3)
- 35 – 2b – b
^{2} - x
^{2}– y^{2}+ ( x + y )^{2} - 25a
^{2}– 4 ( a – 2b )^{2}.

### Solutions.

6a2 + 15x + 9.

Since 3 is a common factor to all the terms, first take out 3 as the common factor:

6a^{2} + 15a + 9 = 3 (2a2+ 5a + 3)

= 3 (2a^{2} + 2a + 3a + 3)

= 3 (2a (a+1) + 3 (a+1)

= 3 ( (a+1) (2a+ 3)).

Hence.

6a^{2} + 15a + 9 = 3 (a+ 1) (2a + 3)

= 3 (a+1) ( 2a + 3)

- 6a
^{2}– 19ax – 36x^{2}

1^{st} step: Find the product of the first and last terms.

6a^{2} x -36x^{2} = -216a^{2}x^{2}

2^{nd} step: Find two terms such that their products is – 216a2x and their sum is -19ax ( the middle term).

Factors of -216a^{2}x^{2} sum of factors .

- +27ax and -8ax + 19ax
- -27ax and +8ax – 19ax
- +9ac and -24ax – 15ax
- -9x and 24ax + 15ax.

Of these only b gives the required result.

3^{rd} step: Replace -19ax in the given expression by -27ax and 8ax then factorize by grouping:

6a^{2} – 19ax – 36x^{2}

= 6a^{2} – 27ax + 8ax – 36x^{2}

= 3a (2a – 9x) + 4x (2a-9x)

=(2a – 9x ) ( 3a + 4x)

hence,

6a^{2} – 19ax – 36x^{2} = (2a -9x) (3a + 4x)

- (5x – 1 (x -3) – (x -5)(x – 3)

= (x – 3) ( 5x -1) – (x – 5)

= ( x -3) ( 5x – 1 – x + 5 )

= ( x – 3) ( 5x – x + 5 – 1)

= ( x – 3) ( 4x + 4)

= ( x – 3) + ( x + 1)

= ( x -3) 4(x+1)

= 4(x-3) (x+1)

- 35 – 2b –b
^{2}

or – b^{2} – 2b + 35

= -b2 – 7b + 5b + 35

= -b (b+7) + 5 (b + 7)

= (b+7) ( -b + 5)

= (b+7) ( 5-b)

or (7+b) ( 5 – b)

- x
^{2}– y^{2}+ (x + y)^{2}

Since

X^{2} – y^{2} = (x)^{2} – (y)^{2}

= ( x + y) ( x –y)

then x^{2} – y^{2} + (x + y)^{2} = (x + y)(x-y) + (x+ y)^{2}

= (x +y)( x-y + (x + y)

= (x + y ( x –y + x + y)

= ( x + y ) ( x + x

= (x + y) (2x)

(x +y ) (2x) = 2x ( x +y)

### EVALUATION (USE THE BOX AT THE BOTTOM TO POST YOUR ANSWER FOR DISCUSSION AND APPRAISAL).

Factorize the following

- m
^{2}– 15mm – 54n^{2} - 8a
^{2}– 18b^{2} - If 17x = 37 5
^{2}– 356^{2}

Find the value of x

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