Solve -7.5-x/x=12.5-x/2

$$- \frac { 7.5 - x } { x } = \frac { 12.5 - x } { 2 }$$

Solve for x

$$x = \frac{\sqrt{681} + 21}{4} \approx 11.773994175$$

$$x=\frac{21-\sqrt{681}}{4}\approx -1.273994175$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of x,2.

$$-2\left(7.5-x\right)=x\left(12.5-x\right)$$

Use the distributive property to multiply -2 by 7.5-x.

$$-15+2x=x\left(12.5-x\right)$$

Use the distributive property to multiply x by 12.5-x.

$$-15+2x=12.5x-x^{2}$$

Subtract 12.5x from both sides.

$$-15+2x-12.5x=-x^{2}$$

Combine 2x and -12.5x to get -10.5x.

$$-15-10.5x=-x^{2}$$

Add x^{2} to both sides.

$$-15-10.5x+x^{2}=0$$

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.

$$x^{2}-10.5x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -10.5 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10.5\right)±\sqrt{\left(-10.5\right)^{2}-4\left(-15\right)}}{2}$$

Square -10.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-10.5\right)±\sqrt{110.25-4\left(-15\right)}}{2}$$

Multiply -4 times -15.

$$x=\frac{-\left(-10.5\right)±\sqrt{110.25+60}}{2}$$

Add 110.25 to 60.

$$x=\frac{-\left(-10.5\right)±\sqrt{170.25}}{2}$$

Take the square root of 170.25.

$$x=\frac{-\left(-10.5\right)±\frac{\sqrt{681}}{2}}{2}$$

The opposite of -10.5 is 10.5.

$$x=\frac{10.5±\frac{\sqrt{681}}{2}}{2}$$

Now solve the equation x=\frac{10.5±\frac{\sqrt{681}}{2}}{2} when ± is plus. Add 10.5 to \frac{\sqrt{681}}{2}.

$$x=\frac{\sqrt{681}+21}{2\times 2}$$

Divide \frac{21+\sqrt{681}}{2} by 2.

$$x=\frac{\sqrt{681}+21}{4}$$

Now solve the equation x=\frac{10.5±\frac{\sqrt{681}}{2}}{2} when ± is minus. Subtract \frac{\sqrt{681}}{2} from 10.5.

$$x=\frac{21-\sqrt{681}}{2\times 2}$$

Divide \frac{21-\sqrt{681}}{2} by 2.

$$x=\frac{21-\sqrt{681}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{681}+21}{4} x=\frac{21-\sqrt{681}}{4}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2x, the least common multiple of x,2.

$$-2\left(7.5-x\right)=x\left(12.5-x\right)$$

Use the distributive property to multiply -2 by 7.5-x.

$$-15+2x=x\left(12.5-x\right)$$

Use the distributive property to multiply x by 12.5-x.

$$-15+2x=12.5x-x^{2}$$

Subtract 12.5x from both sides.

$$-15+2x-12.5x=-x^{2}$$

Combine 2x and -12.5x to get -10.5x.

$$-15-10.5x=-x^{2}$$

Add x^{2} to both sides.

$$-15-10.5x+x^{2}=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$-10.5x+x^{2}=15$$

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.

$$x^{2}-10.5x=15$$

Divide -10.5, the coefficient of the x term, by 2 to get -5.25. Then add the square of -5.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10.5x+\left(-5.25\right)^{2}=15+\left(-5.25\right)^{2}$$

Square -5.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-10.5x+27.5625=15+27.5625$$

Add 15 to 27.5625.

$$x^{2}-10.5x+27.5625=42.5625$$

Factor x^{2}-10.5x+27.5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5.25\right)^{2}=42.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5.25\right)^{2}}=\sqrt{42.5625}$$

Simplify.

$$x-5.25=\frac{\sqrt{681}}{4} x-5.25=-\frac{\sqrt{681}}{4}$$

Add 5.25 to both sides of the equation.

$$x=\frac{\sqrt{681}+21}{4} x=\frac{21-\sqrt{681}}{4}$$