Question

directions: find the

1. simplify

\\( \frac { x ^ { 2 } + x - 6 } { x ^ { 2 } - 6 x - 27 } \\)
(a) if \\( \frac { x + 2 } { x + 9 } \\) rwer is draw the following body around the circle.
(b) if \\( \frac { x - 2 } { x - 9 } \\) ver is draw the following body around the circle.

Explanation:

Step1: Factor numerator and denominator

For the numerator \(x^{2}+x - 6\), we use the formula \(ax^{2}+bx + c\) (\(a = 1\), \(b=1\), \(c=-6\)). We need two numbers \(m\) and \(n\) such that \(m + n=b\) and \(m\times n=ac\). Here, \(m = 3\) and \(n=-2\) since \(3+( - 2)=1\) and \(3\times(-2)=-6\). So \(x^{2}+x - 6=(x + 3)(x - 2)\).

For the denominator \(x^{2}-6x - 27\), \(a = 1\), \(b=-6\), \(c=-27\). We find two numbers \(p\) and \(q\) such that \(p + q=b\) and \(p\times q=ac\). Here, \(p=-9\) and \(q = 3\) since \(-9+3=-6\) and \(-9\times3=-27\). So \(x^{2}-6x - 27=(x - 9)(x + 3)\).

The fraction becomes \(\frac{(x + 3)(x - 2)}{(x - 9)(x + 3)}\).

Step2: Cancel out the common factor

Cancel out the common factor \((x + 3)\) (assuming \(x
eq - 3\)). The simplified fraction is \(\frac{x - 2}{x - 9}\).

Answer:

The simplified form is \(\frac{x - 2}{x - 9}\), so the answer is (b) \(\frac{x - 2}{x - 9}\) (draw the corresponding body around the circle for option (b)).