Solve h(x-3)=-8/4(x-3)^2+5

$$h ( x - 3 ) = \frac { - 8 } { 4 ( x - 3 ) ^ { 2 } } + 5$$

Evaluate

$$\frac{5x^{2}-30x+43}{\left(x-3\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}.

$$\frac{-8}{4\left(x-3\right)^{2}}+\frac{5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}$$

Since \frac{-8}{4\left(x-3\right)^{2}} and \frac{5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-8+5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}$$

Do the multiplications in -8+5\times 4\left(x-3\right)^{2}.

$$\frac{-8+20x^{2}-120x+180}{4\left(x-3\right)^{2}}$$

Combine like terms in -8+20x^{2}-120x+180.

$$\frac{172+20x^{2}-120x}{4\left(x-3\right)^{2}}$$

Factor the expressions that are not already factored in \frac{172+20x^{2}-120x}{4\left(x-3\right)^{2}}.

$$\frac{4\times 5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{4\left(x-3\right)^{2}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{\left(x-3\right)^{2}}$$

Expand \left(x-3\right)^{2}.

$$\frac{5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{x^{2}-6x+9}$$

To find the opposite of -\frac{1}{5}\sqrt{10}+3, find the opposite of each term.

$$\frac{5\left(x+\frac{1}{5}\sqrt{10}-3\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{x^{2}-6x+9}$$

To find the opposite of \frac{1}{5}\sqrt{10}+3, find the opposite of each term.

$$\frac{5\left(x+\frac{1}{5}\sqrt{10}-3\right)\left(x-\frac{1}{5}\sqrt{10}-3\right)}{x^{2}-6x+9}$$

Use the distributive property to multiply 5 by x+\frac{1}{5}\sqrt{10}-3.

$$\frac{\left(5x+\sqrt{10}-15\right)\left(x-\frac{1}{5}\sqrt{10}-3\right)}{x^{2}-6x+9}$$

Use the distributive property to multiply 5x+\sqrt{10}-15 by x-\frac{1}{5}\sqrt{10}-3 and combine like terms.

$$\frac{5x^{2}-30x-\frac{1}{5}\left(\sqrt{10}\right)^{2}+45}{x^{2}-6x+9}$$

The square of \sqrt{10} is 10.

$$\frac{5x^{2}-30x-\frac{1}{5}\times 10+45}{x^{2}-6x+9}$$

Multiply -\frac{1}{5} and 10 to get -2.

$$\frac{5x^{2}-30x-2+45}{x^{2}-6x+9}$$

Add -2 and 45 to get 43.

$$\frac{5x^{2}-30x+43}{x^{2}-6x+9}$$

Expand

$$\frac{5x^{2}-30x+43}{\left(x-3\right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}.

$$\frac{-8}{4\left(x-3\right)^{2}}+\frac{5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}$$

Since \frac{-8}{4\left(x-3\right)^{2}} and \frac{5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{-8+5\times 4\left(x-3\right)^{2}}{4\left(x-3\right)^{2}}$$

Do the multiplications in -8+5\times 4\left(x-3\right)^{2}.

$$\frac{-8+20x^{2}-120x+180}{4\left(x-3\right)^{2}}$$

Combine like terms in -8+20x^{2}-120x+180.

$$\frac{172+20x^{2}-120x}{4\left(x-3\right)^{2}}$$

Factor the expressions that are not already factored in \frac{172+20x^{2}-120x}{4\left(x-3\right)^{2}}.

$$\frac{4\times 5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{4\left(x-3\right)^{2}}$$

Cancel out 4 in both numerator and denominator.

$$\frac{5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{\left(x-3\right)^{2}}$$

Expand \left(x-3\right)^{2}.

$$\frac{5\left(x-\left(-\frac{1}{5}\sqrt{10}+3\right)\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{x^{2}-6x+9}$$

To find the opposite of -\frac{1}{5}\sqrt{10}+3, find the opposite of each term.

$$\frac{5\left(x+\frac{1}{5}\sqrt{10}-3\right)\left(x-\left(\frac{1}{5}\sqrt{10}+3\right)\right)}{x^{2}-6x+9}$$

To find the opposite of \frac{1}{5}\sqrt{10}+3, find the opposite of each term.

$$\frac{5\left(x+\frac{1}{5}\sqrt{10}-3\right)\left(x-\frac{1}{5}\sqrt{10}-3\right)}{x^{2}-6x+9}$$

Use the distributive property to multiply 5 by x+\frac{1}{5}\sqrt{10}-3.

$$\frac{\left(5x+\sqrt{10}-15\right)\left(x-\frac{1}{5}\sqrt{10}-3\right)}{x^{2}-6x+9}$$

Use the distributive property to multiply 5x+\sqrt{10}-15 by x-\frac{1}{5}\sqrt{10}-3 and combine like terms.

$$\frac{5x^{2}-30x-\frac{1}{5}\left(\sqrt{10}\right)^{2}+45}{x^{2}-6x+9}$$

The square of \sqrt{10} is 10.

$$\frac{5x^{2}-30x-\frac{1}{5}\times 10+45}{x^{2}-6x+9}$$

Multiply -\frac{1}{5} and 10 to get -2.

$$\frac{5x^{2}-30x-2+45}{x^{2}-6x+9}$$

Add -2 and 45 to get 43.

$$\frac{5x^{2}-30x+43}{x^{2}-6x+9}$$