Factor: 5(x+6)^2-90
I am able to simplify this expression to 5(x^2+12x+18), but i'm stuck on trying to solve the rest of this. I understand that you can use the quadratic formula, but I want to refrain from doing so as this question should be utilising perfect squares/difference of two squares.
you are correct so far.
Note that the discriminant is 144-72 = 72, which is not a perfect square. So, the roots will not be rational, and the quadratic formula is your best bet.
Or, you can complete the square.
5(x^2+12x+18) = 0
x^2+12x+18 = 0
x^2+12x+36 = 18
(x+6)^2 = 18
x+6 = ±√18
x = -6±√18 = -6±3√2
To factor the expression 5(x+6)^2 - 90 without using the quadratic formula, you can use the concept of perfect squares and difference of two squares.
First, let's simplify the expression by expanding the square term:
5(x+6)^2 - 90
= 5(x^2 + 12x + 36) - 90
= 5x^2 + 60x + 180 - 90
= 5x^2 + 60x + 90
Now, let's try to find common factors among the terms. In this case, we can factor out a common factor of 5:
5(x^2 + 12x + 18)
Next, we want to factor the quadratic expression inside the parentheses, x^2 + 12x + 18. To do this, we need to find two numbers whose product is equal to the constant term (18) and whose sum is equal to the coefficient of the middle term (12).
The factors of 18 are:
1, 18
2, 9
3, 6
None of these factor pairs add up to 12, so the quadratic expression cannot be factored using integer numbers.
Therefore, the final factored form of the expression is:
5(x^2 + 12x + 18)