Question

10.

Answer both a and b for all values of x for which
these expressions are defined.
a) Simplify:
x² + 9x + 20
x² - 16
b) Solve for x:
2
=
x-3
5
÷
x2 +5x
4x - 16

Answer 1

a) Simplify:

x² + 9x + 20

x² - 16

We can factor the numerator and the denominator and simplify the expression:

(x + 5)(x + 4)

(x + 4)(x - 4)

Canceling the common factor of x + 4, we get:

x + 5

x - 4

Therefore, the simplified expression is (x + 5)/(x - 4).

b) Solve for x:

2

x-3

5

÷

x2 +5x

4x - 16

We can simplify the expression on the right side of the division symbol by factoring:

2

x - 3

5

×

1

(x + 1)(x + 4)

Now we can multiply the numerator by the reciprocal of the denominator and simplify:

2(x + 1)(x + 4)

(x - 3)(4x - 16)(5)

Expanding the brackets on the right side, we get:

2(x² + 5x + 4)

5(x² - 2x - 12)

Multiplying out and simplifying, we get:

2x² + 10x + 8

5x² - 10x - 60

Bringing all the terms to one side and simplifying, we get:

3x² - 20x - 68 = 0

Now we can solve for x using the quadratic formula:

x = [20 ± sqrt(20² - 4(3)(-68))] / 2(3)

x = [20 ± sqrt(1424)] / 6

x = [20 ± 8sqrt(22)] / 6

x = (10 ± 4sqrt(22)) / 3

Therefore, the solutions are x = (10 + 4sqrt(22)) / 3 and x = (10 - 4sqrt(22)) / 3.

Explanation: