PlosMathLearning

Quadratic Expressions Quadratic Expressions


The difference between two squares

The expression a2 – b2 is considered as the difference between two squares.

The difference between two squares is given in the form a2 – b2 = (a – b)(a + b).

Example 1: Factorize a2 – 25.

a2 – 25 can be written as (a)2 – (5)2 by applying the difference between two squares.

Hence, a2 – 25 = (a - 5)(a + 5)

Example 2: Factorize 9a2 – 16.

9a2 – 16 can be written as (3a)2 – (4)2 by applying the difference between two squares.

Hence, 9a2 – 16 = (3a - 4)(3a + 4)



Factorizing quadratic expressions in the form a2 + bx + c

Example 1: Factorize the quadratic expression x2 + 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, x2 + 2x + 3x + 6

Step 3: Factorize x2 + 2x + 3x + 6 by taking it in pairs.

x2 + 2x + 3x + 6

x(x + 2) + 3(x + 2)

(x + 3)(x + 2)

Example 2: 3x2 +13x + 4. Note that the coefficient of x2 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 x2 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, 3x2 + x + 12x + 4

Step 4: Factorize 3x2 + x + 12x + 4 by taking it in pairs.

3x2 + x + 12x + 4

x(3x + 1) + 4(3x + 1)

(x + 4)(3x + 1)


Practice Questions

Factorize:

1. x2 – 49

2. x2 – 64

3. m2 – 169

4. x2 – 81

5. x2 – 225

6. x4 – 100

7. x4 – 144

8. x2 – y2

9. m2 – n2

10. x4 – y2

11. x4 – y4

12. x2 – y6

13. x6 – y6

14. x10 – y2

15. x10 – y6

16. x2 + 10x + 16

17. x2 + 5x - 6

18. x2 - 7x - 18

19. 2x2 + 8x + 6

20. 5x2 - 8x + 3


Answers

1. x2 – 49

= (x)2 - (7)2

= (x - 7)(x + 7)

2. x2 – 64

= (x)2 - (8)2

= (x - 8)(x + 8)

3. m2 – 169

= (x)2 - (13)2

= (x - 13)(x + 13)

4. x2 – 81

= (x)2 - (9)2

= (x - 9)(x + 9)

5. x2 – 225

= (x)2 - (15)2

= (x - 15)(x + 15)

6. x4 – 100

= (x2)2 - (10)2

= (x2 - 10)(x2 + 10)

7. x4 – 144

= (x2)2 - (12)2

= (x2 - 12)(x2 + 12)

8. x2 – y2

= (x)2 - (15)2

= (x - y)(x + y)

9. m2 – n2

= (m)2 - (n)2

= (m - n)(m + n)

10. x4 – y2

= (x2)2 - (y)2

= (x2 - y)(x2 + y)

11. x4 – y4

= (x2)2 - (y2)2

= (x2 - y2)(x2 + y2)

12. x2 – y6

= (x)2 - (y3)2

= (x - y3)(x + y3)

13. x6 – y6

= (x3)2 - (y3)2

= (x3 - y3)(x3 + y3)

14. x10 – y2

= (x5)2 - (y)2

= (x5 - y)(x5 + y)

15. x10 – y6

= (x5)2 - (y3)2

= (x5 - y3)(x5 + y3)

16. x2 + 10x + 16

= (x + 2)(x + 8)

17. x2 + 5x - 6

= (x - 1)(x + 6)

18. x2 - 7x - 18

= (x - 9)(x + 2)

19. 2x2 + 8x + 6

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

20. 5x2 - 8x + 3

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




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