algebra problem

Question

37.For what value or values of x is the product of 2x + 3 and x − 5 twenty-seven more than x?

Answer 1

37.For what value or values of x is the product of 2x + 3 and x − 5 twenty-seven more than x?

Let's first transform this wordy problem into math shorthand:

We want to find x for the following situation:

(2x+3)*(x-5) = 27+x

Let's then expand out the multiplied portions:

2x²-10x+3x-15 = 27+x

Then, lets group like terms:

2x²--8x-42=0

Now, let's divide by 2 because it is a common factor:

x²-4x-21=0

Now, let's factor:

(x-7)(x+3)=0 (make sure you check your work here!)

Now, for the above statement to be true, we just need x-7 OR x+3 to equal 0 since:

0*(x+3)=0

and

(x-7)*0=0

This means x can be 7 (since 7-7 = 0) or x can be -3 (since -3+3 = 0)

Answer 2

(2x+3)(x-5) = x + 27

2x^2 - 10x + 3x - 15 = x + 27

2x^2 - 7x - 15 = x + 27

2x^2 - 8x - 42 = 0

x^2 - 4x - 21 = 0

(x - 7)(x+3) = 0

x = 7 or x = -3

( 2*7+3)(7-5) = 17*2 = 34

= 27 + 7

YES

( 2 * -3 + 3)(-3 - 5) = -3 * -8 = 24

= 27 + -3

YES

x = 7 or x = -3