Question

plifying the quadratic formula and transformations

1. solve in simplest radical form: $x = \frac{-7}{x - 6}$

by quadratic formula method:

Explanation:

Step1: Cross - multiply

Multiply both sides by $x - 6$ to get a quadratic equation: $x(x - 6)=-7$.
Expand to $x^{2}-6x + 7=0$.

Step2: Identify coefficients

For the quadratic equation $ax^{2}+bx + c = 0$, here $a = 1$, $b=-6$, $c = 7$.

Step3: Apply quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$.
Substitute the values: $x=\frac{6\pm\sqrt{(-6)^{2}-4\times1\times7}}{2\times1}$.

Step4: Simplify the expression

First, calculate inside the square - root: $(-6)^{2}-4\times1\times7=36 - 28 = 8$.
So $x=\frac{6\pm\sqrt{8}}{2}=\frac{6\pm2\sqrt{2}}{2}=3\pm\sqrt{2}$.

Answer:

$x = 3+\sqrt{2},x = 3-\sqrt{2}$