Factorize these expressions using quadratic trinomials
P²+22p+96,x²+18x-63
which pair of factors of 96 add up to 22? (p+6)(p+16)
which pair of factors of 63 differ by 18? (q+21)(q-3)
To factorize quadratic trinomials, we need to find two binomials that, when multiplied, will give us the quadratic trinomial in question.
Let's start with the expression P² + 22P + 96.
Step 1: Identify the factors of the constant term (96).
The factors of 96 are:
1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Step 2: Find a combination of factors whose sum is equal to the coefficient of the linear term (22).
In this case, the combination that gives us a sum of 22 is 4 and 24 (4 + 24 = 28).
Step 3: Rewrite the quadratic expression using the factors found in step 2.
P² + 4P + 24P + 96
Step 4: Group the terms and factor by grouping.
(P² + 4P) + (24P + 96)
P(P + 4) + 24(P + 4)
Step 5: Factor out the common binomial.
(P + 4)(P + 24)
Therefore, the factorization of P² + 22P + 96 is (P + 4)(P + 24).
Now let's move on to the expression x² + 18x - 63.
Step 1: Identify the factors of the constant term (-63).
The factors of -63 are:
-1, 1, -3, 3, -7, 7, -9, 9, -21, 21, -63, 63
Step 2: Find a combination of factors whose sum is equal to the coefficient of the linear term (18).
In this case, the combination that gives us a sum of 18 is 21 and -3 (21 + (-3) = 18).
Step 3: Rewrite the quadratic expression using the factors found in step 2.
x² + 21x - 3x - 63
Step 4: Group the terms and factor by grouping.
(x² + 21x) + (-3x - 63)
x(x + 21) - 3(x + 21)
Step 5: Factor out the common binomial.
(x - 3)(x + 21)
Therefore, the factorization of x² + 18x - 63 is (x - 3)(x + 21).