# How do you simplify (x^2 - 7x + 10)/( x^2 - 10x + 25) * (x - 5)/( x - 2)?

Jun 18, 2018

$1$

#### Explanation:

In questions on algebraic fractions - the first step is to factorise as far as possible.

$\frac{{x}^{2} - 7 x + 10}{{x}^{2} - 10 x + 25} \cdot \frac{x - 5}{x - 2}$

$= \frac{\left(x - 5\right) \left(x - 2\right)}{\left(x - 5\right) \left(x - 5\right)} \cdot \frac{\left(x - 5\right)}{\left(x - 2\right)}$

If you have factors which are all multipied together, then you can cancel like factors.

$= \frac{\cancel{\left(x - 5\right)} \cancel{\left(x - 2\right)}}{\cancel{\left(x - 5\right)} \cancel{\left(x - 5\right)}} \cdot \frac{\cancel{\left(x - 5\right)}}{\cancel{\left(x - 2\right)}}$

$= 1$

Jun 18, 2018

$\frac{{x}^{2} - 7 x + 10}{{x}^{2} - 10 x + 25}$.$\frac{x - 5}{x - 2}$ = $1$

#### Explanation:

Factorize first:

Step 1: Factorize ${x}^{2} - 7 x + 10$

Lets find two numbers or factors, when multiplied, gives $10$ and when added together it gives $- 7$

$2 \times 5 = 10$

$2 \times - 5 = - 10$

$- 2 \times - 5 = 10$ -----> This is the one!

Re-write the equation:

${x}^{2} - 2 x - 5 x + 10$

$x \left(x - 2\right) - 5 \left(x - 2\right)$

$\left(x - 2\right) \left(x - 5\right)$

Step 2: Factorize ${x}^{2} - 10 x + 25$

Lets find two numbers or factors, when multiplied, gives $25$ and when added together it gives $- 10$

$5 \times 5 = 25$

$5 \times - 5 = - 25$

$- 5 \times - 5 = 25$ -----> This is the one!

Re-write the equation:

${x}^{2} - 5 x - 5 x + 25$

$x \left(x - 5\right) - 5 \left(x - 5\right)$

$\left(x - 5\right) \left(x - 5\right)$

Lets write the whole equation with the factors we got above and then simplify further.

$\frac{{x}^{2} - 7 x + 10}{{x}^{2} - 10 x + 25}$.$\frac{x - 5}{x - 2}$ = $\frac{\left(x - 2\right) \left(x - 5\right)}{\left(x - 5\right) \left(x - 5\right)}$.$\frac{x - 5}{x - 2}$

$\frac{\cancel{x - 2} \cancel{x - 5}}{\cancel{x - 5} \cancel{x - 5}}$.$\frac{\cancel{x - 5}}{\cancel{x - 2}}$ = $1$

Answer is $1$