# Using complete sentences, explain each step in simplifying the ratio of [(x2)+ 8x]/[x2 + (10x) + 16]?i cant seem to get this!

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To simplify the ratio, we'll focus on the numerator and denominator, separately.

We notice that the denominator is a quadratic and we'll apply the quadratic formula to calculate it's roots.

The quadratic formula is:

x1 = [-b+sqrt(b^2 - 4ac)]/2a

x2 = [-b-sqrt(b^2 - 4ac)]/2a

a,b,c, are the coefficients of the quadratic: ax^2 + bx + c.

We'll identify a,b,c:

a = 1

b = 10

c = 16

We'll substitute them into the formula:

x1 = [-b+sqrt(b^2 - 4ac)]/2a

x1 = [-10+sqrt(100 - 64)]/2

x1 = (-10+sqrt36)/2

x1 = (-10+6)/2

x1 = -2

x2 = (-10-6)/2

x2 = -8

Now, we'll write the quadratic:

a(x - x1)(x - x2) = 1*(x+2)(x+8)

The denominator will become:

x^2 + 10x + 16 = (x+2)(x+8)

We notice that we can factorize the numerator, by x:

x^2 + 8x = x(x+8)

We'll re-write the ratio:

(x^2 + 8x)/(x^2 + 10x + 16) = x(x+8)/(x+2)(x+8)

We'll reduce like terms, namely (x+8):

**(x^2 + 8x)/(x^2 + 10x + 16) = x/(x+2)**

To simplify the given ratio [ x^2 + 8x] / [ x^2 + 10x +16] follow these steps:

- Find the factors of the numerator and the denominator:

=> x^2 + 8x = x*(x+8)

=> x^2 + 10x +16 = x^2+ 8x + 2x + 16= x (x+8) + 2(x+8) = (x+2)(x+8)

So we have the ratio as:

[x*(x+8)] / [ (x+2)(x+8)]

- Now cancel the common factors:

=> [x*(x+8)] / [ (x+2)(x+8)]

=> x / (x+2)

**The required result is x / (x+2)**

To simplify the rational expression [x^2+8x]/[x^2+10x+16], we take the numerator and show it as a prduct of two factors. Then we take denominator and show it as product of two factors. We examine for any common factor between numerator and denominator. Then we divide (or reduce or cancel) both numerator and denominator by the common factor. That completes the simplification of the rational expression.

Numerator, x^2 +8x = x(x+8).

Denominator , x^2+10x+16 = x^2+8x+2x+16 .

x^2+10x+16= x(x+8)+2(x+8).

x^2+10x+16 = (x+8)(x+2).

Therefore, the rational expression, (x^2+8x)/(x^2+10x+16) = x(x+8)/{(x+8)(x+2)}. Now we find that x+8 is a common factor of both numerator and denominator of the rational expression on left. So we divide both numerator and denominator by x+8 and rewrite the given rational expression and its equivalent rational expression in the simplified form:

(x^2+8x)/(x^2+10x+16) = x/(x+2).

Hope this helps.