Factor each of the following quadratic expressions.
Question 1 :
x^{2} + 6x + 8
Question 2 :
x^{2} + 9x + 18
Question 3 :
x^{2} + 3x -18
Question 4 :
x^{2} - 2x - 15
Question 5 :
x^{2} - 4x + 3
Question 6 :
3x^{2} + 10x + 8
Question 7 :
3x^{2} – 5x – 12
Question 8 :
x^{2} – 9
Question 9 :
9x^{2} - 49
Question 10 :
(x + 2)^{2} - 25
1. Answer :
= x^{2} + 6x + 8
Multiply the coefficient x^{2}, 1 and the constant term 8.
= 1 x 8
= 8
Find two numbers such that the product is equal to 8 and the sum is equal to the coeffient of x, 6.
The two numbers satisfy the above condition are 2 and 4.
Split the middle term 6x using the two numbers 2 and 4.
= x^{2} + 2x + 4x + 8
Factor by grouping.
= (x^{2} + 2x) + (4x + 8)
= x(x + 2) + 4(x + 2)
= (x + 2)(x + 4)
2. Answer :
= x^{2} + 9x + 18
Multiply the coefficient x^{2}, 1 and the constant term 18.
= 1 x 18
= 18
Find two numbers such that the product is equal to 18 and the sum is equal to the coeffient of x, 9.
The two numbers satisfy the above condition are 3 and 6.
Split the middle term 9x using the two numbers 3 and 6.
= x^{2} + 3x + 6x + 18
Factor by grouping.
= (x^{2} + 3x) + (6x + 18)
= x(x + 3) + 6(x + 3)
= (x + 3)(x + 6)
3. Answer :
= x^{2} + 3x - 18
Multiply the coefficient x^{2}, 1 and the constant term -18.
= 1 x (-18)
= -18
Find two numbers such that the product is equal to -18 and the sum is equal to the coeffient of x, 3.
The two numbers satisfy the above condition are -3 and 6.
Split the middle term 3x using the two numbers -3 and 6.
= x^{2} - 3x + 6x - 18
Factor by grouping.
= (x^{2} - 3x) + (6x - 18)
= x(x - 3) + 6(x - 3)
= (x - 3)(x + 6)
4. Answer :
= x^{2} - 2x - 15
Multiply the coefficient x^{2}, 1 and the constant term -15.
= 1 x (-15)
= -15
Find two numbers such that the product is equal to -15 and the sum is equal to the coeffient of x, -2.
The two numbers satisfy the above condition are -5 and 3.
Split the middle term -2x using the two numbers -5 and 3.
= x^{2} - 5x + 3x - 15
Factor by grouping.
= (x^{2} - 5x) + (3x - 15)
= x(x - 5) + 3(x - 5)
= (x - 5)(x + 3)
5. Answer :
= x^{2} - 4x + 3
Multiply the coefficient x^{2}, 1 and the constant term 3.
= 1 x 3
= 3
Find two numbers such that the product is equal to 3 and the sum is equal to the coeffient of x, -4.
The two numbers satisfy the above condition are -3 and -1.
Split the middle term -4x using the two numbers -3 and -1.
= x^{2} - 3x - x + 3
Factor by grouping.
= (x^{2} - 3x) + (-x + 3)
= x(x - 3) - 1(x - 3)
= (x - 3)(x - 1)
6. Answer :
= 3x^{2} + 10x + 8
Multiply the coefficient x^{2}, 3 and the constant term 8.
= 3 x 8
= 24
Find two numbers such that the product is equal to 24 and the sum is equal to the coeffient of x, 10.
The two numbers satisfy the above condition are 3 and 8.
Split the middle term 10x using the two numbers 3 and 8.
= 3x^{2} + 3x + 8x + 8
Factor by grouping.
= (3x^{2} + 3x) + (8x + 8)
= 3x(x + 1) + 8(x + 1)
= (x + 1)(3x + 8)
7. Answer :
= 3x^{2} – 5x – 12
Multiply the coefficient x^{2}, 3 and the constant term -12.
= 3 x (-12)
= -36
Find two numbers such that the product is equal to -36 and the sum is equal to the coeffient of x, -5.
The two numbers satisfy the above condition are -9 and 4.
Split the middle term -5x using the two numbers -9 and 4.
= 3x^{2} - 9x + 4x - 12
Factor by grouping.
= (3x^{2} - 9x) + (4x - 12)
= 3x(x - 3) + 4(x - 3)
= (x - 3)(3x + 4)
8. Answer :
The quadratic expression x^{2} – 9 can be factored using the following algebraic identity.
a^{2} - b^{2} = (a + b)(a - b)
= x^{2} – 9
= x^{2} – 3^{2}
= (x + 3)(x - 3)
9. Answer :
= 9x^{2} - 49
= 3^{2}x^{2} – 7^{2}
= (3x)^{2} – 7^{2}
= (3x + 7)(3x - 7)
10. Answer :
= (x + 2)^{2} - 25
= (x + 2)^{2} – 5^{2}
= (x + 2 + 5)(x + 2 - 5)
= (x + 7)(x - 3)
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