Let's start by determining the restrictions and then simplify the expression.

Given Expression:

$\frac{25x^2 - 80x + 63}{5x^2 - 47x + 56}$

Step 1: Find the Restrictions

The restrictions are values of $x$ that make the denominator equal to zero. We need to solve the quadratic equation in the denominator:

$5x^2 - 47x + 56 = 0$

Let's solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 5$, $b = -47$, and $c = 56$.

$x = \frac{47 \pm \sqrt{(-47)^2 - 4(5)(56)}}{2(5)}$ $x = \frac{47 \pm \sqrt{2209 - 1120}}{10}$ $x = \frac{47 \pm \sqrt{1089}}{10}$ $x = \frac{47 \pm 33}{10}$

So, the roots are: $x_1 = \frac{47 + 33}{10} = \frac{80}{10} = 8$ $x_2 = \frac{47 - 33}{10} = \frac{14}{10} = 1.4$

Restrictions: $x \neq 8 \quad \text{and} \quad x \neq 1.4$

Step 2: Simplify the Expression

Now, we simplify the given expression by factoring both the numerator and the denominator.

Numerator: $25x^2 - 80x + 63$ Factor this quadratic expression: $25x^2 - 80x + 63 = (5x - 7)(5x - 9)$

Denominator: $5x^2 - 47x + 56$ Factor this quadratic expression: $5x^2 - 47x + 56 = (5x - 7)(x - 8)$

Now, substitute the factors back into the original expression: $\frac{(5x - 7)(5x - 9)}{(5x - 7)(x - 8)}$

Cancel the common factor $5x - 7$ from both the numerator and the denominator: $\frac{5x - 9}{x - 8}$

Final Answer:

• Restrictions: $x \neq 8 \quad \text{and} \quad x \neq 1.4$

• Simplified Expression: $\frac{5x - 9}{x - 8}$

Would you like more details, or do you have any questions?

Here are 5 related questions you might find useful:

1. How do you factor a quadratic expression?
2. What is the quadratic formula, and when is it used?
3. What are the steps to simplify rational expressions?
4. Can you provide more examples of finding restrictions in rational expressions?
5. How do you check if the factors obtained are correct?

Tip: Always check for common factors before simplifying to avoid mistakes in cancellation.