Solve 75/8x+75/8(x-10)=1

$$\frac{ 75 }{ 8x } + \frac{ 75 }{ 8(x-10) } =1$$

Solve for x

$$x=25$$

$$x = \frac{15}{4} = 3\frac{3}{4} = 3.75$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 8x\left(x-10\right), the least common multiple of 8x,8\left(x-10\right).

$$\left(x-10\right)\times 75+x\times 75=8x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 75.

$$75x-750+x\times 75=8x\left(x-10\right)$$

Combine 75x and x\times 75 to get 150x.

$$150x-750=8x\left(x-10\right)$$

Use the distributive property to multiply 8x by x-10.

$$150x-750=8x^{2}-80x$$

Subtract 8x^{2} from both sides.

$$150x-750-8x^{2}=-80x$$

Add 80x to both sides.

$$150x-750-8x^{2}+80x=0$$

Combine 150x and 80x to get 230x.

$$230x-750-8x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-8x^{2}+230x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 230 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-230±\sqrt{230^{2}-4\left(-8\right)\left(-750\right)}}{2\left(-8\right)}$$

Square 230.

$$x=\frac{-230±\sqrt{52900-4\left(-8\right)\left(-750\right)}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-230±\sqrt{52900+32\left(-750\right)}}{2\left(-8\right)}$$

Multiply 32 times -750.

$$x=\frac{-230±\sqrt{52900-24000}}{2\left(-8\right)}$$

Add 52900 to -24000.

$$x=\frac{-230±\sqrt{28900}}{2\left(-8\right)}$$

Take the square root of 28900.

$$x=\frac{-230±170}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-230±170}{-16}$$

Now solve the equation x=\frac{-230±170}{-16} when ± is plus. Add -230 to 170.

$$x=-\frac{60}{-16}$$

Reduce the fraction \frac{-60}{-16} to lowest terms by extracting and canceling out 4.

$$x=\frac{15}{4}$$

Now solve the equation x=\frac{-230±170}{-16} when ± is minus. Subtract 170 from -230.

$$x=-\frac{400}{-16}$$

Divide -400 by -16.

$$x=25$$

The equation is now solved.

$$x=\frac{15}{4} x=25$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,10 since division by zero is not defined. Multiply both sides of the equation by 8x\left(x-10\right), the least common multiple of 8x,8\left(x-10\right).

$$\left(x-10\right)\times 75+x\times 75=8x\left(x-10\right)$$

Use the distributive property to multiply x-10 by 75.

$$75x-750+x\times 75=8x\left(x-10\right)$$

Combine 75x and x\times 75 to get 150x.

$$150x-750=8x\left(x-10\right)$$

Use the distributive property to multiply 8x by x-10.

$$150x-750=8x^{2}-80x$$

Subtract 8x^{2} from both sides.

$$150x-750-8x^{2}=-80x$$

Add 80x to both sides.

$$150x-750-8x^{2}+80x=0$$

Combine 150x and 80x to get 230x.

$$230x-750-8x^{2}=0$$

Add 750 to both sides. Anything plus zero gives itself.

$$230x-8x^{2}=750$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}+230x=750$$

Divide both sides by -8.

$$\frac{-8x^{2}+230x}{-8}=\frac{750}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{230}{-8}x=\frac{750}{-8}$$

Reduce the fraction \frac{230}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{115}{4}x=\frac{750}{-8}$$

Reduce the fraction \frac{750}{-8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{115}{4}x=-\frac{375}{4}$$

Divide -\frac{115}{4}, the coefficient of the x term, by 2 to get -\frac{115}{8}. Then add the square of -\frac{115}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{115}{4}x+\left(-\frac{115}{8}\right)^{2}=-\frac{375}{4}+\left(-\frac{115}{8}\right)^{2}$$

Square -\frac{115}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{115}{4}x+\frac{13225}{64}=-\frac{375}{4}+\frac{13225}{64}$$

Add -\frac{375}{4} to \frac{13225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{115}{4}x+\frac{13225}{64}=\frac{7225}{64}$$

Factor x^{2}-\frac{115}{4}x+\frac{13225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{115}{8}\right)^{2}=\frac{7225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{115}{8}\right)^{2}}=\sqrt{\frac{7225}{64}}$$

Simplify.

$$x-\frac{115}{8}=\frac{85}{8} x-\frac{115}{8}=-\frac{85}{8}$$

Add \frac{115}{8} to both sides of the equation.

$$x=25 x=\frac{15}{4}$$