Question $\frac{x^{2}-5 x}{2 x^{2}-a^{2}} \div \frac{x^{2}-7 x+10}{2 x^{2}-5 a+2}$

Studdy Solution

STEP 1

Assumptions
1. The given expression is $\frac{x^{2}-5 x}{2 x^{2}-a^{2}} \div \frac{x^{2}-7 x+10}{2 x^{2}-5 a+2}$ .
2. We need to simplify the given expression.
3. We assume $x \neq 0$ and $a \neq 0$ to avoid division by zero.

STEP 2

First, rewrite the division as a multiplication by the reciprocal.
$\frac{x^{2}-5 x}{2 x^{2}-a^{2}} \div \frac{x^{2}-7 x+10}{2 x^{2}-5 a+2} = \frac{x^{2}-5 x}{2 x^{2}-a^{2}} \times \frac{2 x^{2}-5 a+2}{x^{2}-7 x+10}$

STEP 3

Factorize the numerators and denominators where possible. Start with the numerator $x^{2} - 5x$ .
$x^{2} - 5x = x(x - 5)$

STEP 4

Factorize the denominator $2x^{2} - a^{2}$ using the difference of squares.
$2x^{2} - a^{2} = ( \sqrt{2}x - a)( \sqrt{2}x + a)$

STEP 5

Factorize the numerator $x^{2} - 7x + 10$ .
$x^{2} - 7x + 10 = (x - 2)(x - 5)$

STEP 6

Factorize the denominator $2x^{2} - 5a + 2$ . This expression does not factorize easily, so we leave it as is.
$2x^{2} - 5a + 2$

STEP 7

Substitute the factored forms back into the expression.
$\frac{x(x - 5)}{(\sqrt{2}x - a)(\sqrt{2}x + a)} \times \frac{2x^{2} - 5a + 2}{(x - 2)(x - 5)}$

STEP 8

Cancel out the common factors in the numerators and denominators. Here, $x - 5$ is a common factor.
$\frac{x}{(\sqrt{2}x - a)(\sqrt{2}x + a)} \times \frac{2x^{2} - 5a + 2}{x - 2}$

STEP 9

Combine the remaining fractions.
$\frac{x (2x^{2} - 5a + 2)}{(\sqrt{2}x - a)(\sqrt{2}x + a)(x - 2)}$
This is the simplified form of the given expression.

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