Example 1 :
Factor the following :
(i) x^{2} + 10x + 24
Solution :
In the quadratic expression above, the coefficient of x^{2} is 1.
Decompose the constant term +24 into two factors such that the product of the two factors is equal to +24 and the addition of two factors is equal to the coefficient of x, that is +10.
Then, the two factors of +24 are
+4 and +6
Factor the given quadratic expression using +4 and +6.
x^{2} + 10x + 24 = (x + 4)(x + 6)
Therefore, the factors of the given quadratic expression are
(x + 4) and (x + 6)
(ii) z^{2} + 4z - 12
Solution :
In the quadratic expression above, the coefficient of z^{2} is 1.
Decompose the constant term -12 into two factors such that the product of the two factors is equal to -12 and the addition of two factors is equal to the coefficient of z, that is +4.
Then, the two factors of -12 are
-2 and +6
Factor the given quadratic expression using -2 and +6.
z^{2} + 4z - 12 = (z - 2)(z + 6)
Therefore, the factors of the given quadratic expression are
(z - 2) and (z + 6)
(iii) p^{2} - 6p - 16
Solution :
In the quadratic expression above, the coefficient of p^{2} is 1.
Decompose the constant term -16 into two factors such that the product of the two factors is equal to -16 and the addition of two factors is equal to the coefficient of p, that is -6.
Then, the two factors of -16 are
-8 and +2
Factor the given quadratic expression using -8 and +2.
p^{2} - 6p - 16 = (p - 8)(p + 2)
Therefore, the factors of the given quadratic expression are
(p - 8) and (p + 2)
(iv) t^{2} - 17t + 72
Solution :
In the quadratic expression above, the coefficient of t^{2} is 1.
Decompose the constant term +72 into two factors such that the product of the two factors is equal to +72 and the addition of two factors is equal to the coefficient of t, that is -17.
Then, the two factors of +72 are
-8 and -9
Factor the given quadratic expression using -8 and -9.
t^{2} - 17t + 72 = (t - 8)(t - 9)
Therefore, the factors of the given quadratic expression are
(t - 8) and (t - 9)
(v) y^{2} - 16y - 80
Solution :
In the quadratic expression above, the coefficient of y^{2} is 1.
Decompose the constant term -80 into two factors such that the product of the two factors is equal to -80 and the addition of two factors is equal to the coefficient of y, that is -16.
Then, the two factors of -80 are
-20 and +4
Factor the given quadratic expression using -20 and +4.
y^{2} - 16y - 80 = (y - 20)(y + 4)
Therefore, the factors of the given quadratic expression are
(y - 20) and (y + 4)
(vi) a^{2} + 10a - 600
Solution :
In the quadratic expression above, the coefficient of a^{2} is 1.
Decompose the constant term -600 into two factors such that the product of the two factors is equal to -600 and the addition of two factors is equal to the coefficient of a, that is +10.
Then, the two factors of -600 are
+30 and -20
Factor the given quadratic expression using +30 and -20.
a^{2} + 10a - 600 = (a + 30)(a - 20)
Therefore, the factors of the given quadratic expression are
(a + 30) and (a - 20)
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