The expression a

**Quadratic Expressions****The difference between two squares**The expression a

^{2}– b^{2}is considered as the difference between two squares. The difference between two squares is given in the form a^{2}– b^{2}= (a – b)(a + b). Example 1: Factorize a^{2}– 25. a^{2}– 25 can be written as (a)^{2}– (5)^{2}by applying the difference between two squares. Hence, a^{2}– 25 = (a - 5)(a + 5) Example 2: Factorize 9a^{2}– 16. 9a^{2}– 16 can be written as (3a)^{2}– (4)^{2}by applying the difference between two squares. Hence, 9a^{2}– 16 = (3a - 4)(3a + 4)**Factorizing quadratic expressions in the form a**^{2}+ bx + c__Example 1:__Factorize the quadratic expression x^{2}+ 5x + 6 Step 1: Find two numbers which when multiplied together equals to the constant (6) and which when add together equals to the coefficient of x (5).**The numbers are 2 and 3. CHECK: 2 x 3 = 6 and 2 + 3 = 5**Step 2: Use the numbers 2 and 3 to replace 5x with 2x + 3x. So we have, x^{2}+ 2x + 3x + 6 Step 3: Factorize x^{2}+ 2x + 3x + 6 by taking it in pairs. x^{2}+ 2x + 3x + 6 x(x + 2) + 3(x + 2) (x + 3)(x + 2)__Example 2:__3x^{2}+13x + 4. Note that the coefficient of x^{2}is more than one. As a result, in this solving this equation you go one step further than in the first example. Step 1: Use the coefficient of x^{2}to multiply the constant. So we have, 3 x 4 = 12. Step 2: Find two numbers which when multiplied together equals 12 and which when added equals 13.**The numbers are 1 and 12. CHECK: 1 x 12 = 12 and 1 + 12 = 13**Step 3: Use the numbers 1 and 12 to rewrite 13x as x + 12x. So we have, 3x^{2}+ x + 12x + 4 Step 4: Factorize 3x^{2}+ x + 12x + 4 by taking it in pairs. 3x^{2}+ x + 12x + 4 x(3x + 1) + 4(3x + 1) (x + 4)(3x + 1)

Practice Questions

Factorize:

**1.** x^{2} – 49

**2.** x^{2} – 64

**3.** m^{2} – 169

**4.** x^{2} – 81

**5.** x^{2} – 225

**6.** x^{4} – 100

**7.** x^{4} – 144

**8.** x^{2} – y^{2}

**9.** m^{2} – n^{2}

**10.** x^{4} – y^{2}

**11.** x^{4} – y^{4}

**12.** x^{2} – y^{6}

**13.** x^{6} – y^{6}

**14.** x^{10} – y^{2}

**15.** x^{10} – y^{6}

**16.** x^{2} + 10x + 16

**17.** x^{2} + 5x - 6

**18.** x^{2} - 7x - 18

**19.** 2x^{2} + 8x + 6

**20.** 5x^{2} - 8x + 3

Answers

**1.** x^{2} – 49

= (x)^{2} - (7)^{2}

= (x - 7)(x + 7)

**2.** x^{2} – 64

= (x)^{2} - (8)^{2}

= (x - 8)(x + 8)

**3.** m^{2} – 169

= (x)^{2} - (13)^{2}

= (x - 13)(x + 13)

**4.** x^{2} – 81

= (x)^{2} - (9)^{2}

= (x - 9)(x + 9)

**5.** x^{2} – 225

= (x)^{2} - (15)^{2}

= (x - 15)(x + 15)

**6.** x^{4} – 100

= (x^{2})^{2} - (10)^{2}

= (x^{2} - 10)(x^{2} + 10)

**7.** x^{4} – 144

= (x^{2})^{2} - (12)^{2}

= (x^{2} - 12)(x^{2} + 12)

**8.** x^{2} – y^{2}

= (x)^{2} - (15)^{2}

= (x - y)(x + y)

**9.** m^{2} – n^{2}

= (m)^{2} - (n)^{2}

= (m - n)(m + n)

**10.** x^{4} – y^{2}

= (x^{2})^{2} - (y)^{2}

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

**11.** x^{4} – y^{4}

= (x^{2})^{2} - (y^{2})^{2}

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

**12.** x^{2} – y^{6}

= (x)^{2} - (y^{3})^{2}

= (x - y^{3})(x + y^{3})

**13.** x^{6} – y^{6}

= (x^{3})^{2} - (y^{3})^{2}

= (x^{3} - y^{3})(x^{3} + y^{3})

**14.** x^{10} – y^{2}

= (x^{5})^{2} - (y)^{2}

= (x^{5} - y)(x^{5} + y)

**15.** x^{10} – y^{6}

= (x^{5})^{2} - (y^{3})^{2}

= (x^{5} - y^{3})(x^{5} + y^{3})

**16.** x^{2} + 10x + 16

= (x + 2)(x + 8)

**17.** x^{2} + 5x - 6

= (x - 1)(x + 6)

**18.** x^{2} - 7x - 18

= (x - 9)(x + 2)

**19.** 2x^{2} + 8x + 6

= (x + 1)(2x + 6)

**20.** 5x^{2} - 8x + 3

= (5x - 3)(x - 1)

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