Determine the restrictions of the expression 1.x/x-2 2.x²+4x-5/3x²+8x-35 3.x²-8/x²+16x+48

To determine the restrictions on each expression, we need to identify the values of \( x \) that would make the denominator zero, as division by zero is undefined.

### Expression 1: \( \frac{x}{x-2} \)

The restriction occurs when the denominator is zero:

\[
x - 2 = 0
\]

\[
x = 2
\]

**Restriction:** \( x \neq 2 \)

### Expression 2: \( \frac{x^2 + 4x - 5}{3x^2 + 8x - 35} \)

To find the restrictions, set the denominator equal to zero:

\[
3x^2 + 8x - 35 = 0
\]

We solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 8 \), and \( c = -35 \).

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(3)(-35)}}{2(3)}
\]

\[
x = \frac{-8 \pm \sqrt{64 + 420}}{6}
\]

\[
x = \frac{-8 \pm \sqrt{484}}{6}
\]

\[
x = \frac{-8 \pm 22}{6}
\]

This gives two solutions:

\[
x = \frac{14}{6} = \frac{7}{3} \quad \text{and} \quad x = \frac{-30}{6} = -5
\]

**Restrictions:** \( x \neq \frac{7}{3} \) and \( x \neq -5 \)

### Expression 3: \( \frac{x^2 - 8}{x^2 + 16x + 48} \)

Again, set the denominator equal to zero:

\[
x^2 + 16x + 48 = 0
\]

Solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 16

Learn how to determine restrictions on the values of x in rational expressions such as x/x-2, (x² + 4x - 5)/(3x² + 8x - 35), and (x² - 8)/(x² + 16x + 48). This problem involves key algebraic concepts, including quadratic equations and the quadratic formula. Suitable for students in Grades 9-12.

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