Solve 1/2/x+1/2/15-x=frac{3}{20}

$$\frac { \frac { 1 } { 2 } } { x } + \frac { \frac { 1 } { 2 } } { 15 - x } = \frac { 3 } { 20 }$$

Solve for x

$$x=5$$

$$x=10$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,15 since division by zero is not defined. Multiply both sides of the equation by 20x\left(x-15\right), the least common multiple of x,15-x,20.

$$\left(20x-300\right)\times \frac{1}{2}-20x\times \frac{1}{2}=3x\left(x-15\right)$$

Use the distributive property to multiply 20x-300 by \frac{1}{2}.

$$10x-150-20x\times \frac{1}{2}=3x\left(x-15\right)$$

Multiply -20 and \frac{1}{2} to get -10.

$$10x-150-10x=3x\left(x-15\right)$$

Combine 10x and -10x to get 0.

$$-150=3x\left(x-15\right)$$

Use the distributive property to multiply 3x by x-15.

$$-150=3x^{2}-45x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}-45x=-150$$

Add 150 to both sides.

$$3x^{2}-45x+150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -45 for b, and 150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 3\times 150}}{2\times 3}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\times 3\times 150}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-45\right)±\sqrt{2025-12\times 150}}{2\times 3}$$

Multiply -12 times 150.

$$x=\frac{-\left(-45\right)±\sqrt{2025-1800}}{2\times 3}$$

Add 2025 to -1800.

$$x=\frac{-\left(-45\right)±\sqrt{225}}{2\times 3}$$

Take the square root of 225.

$$x=\frac{-\left(-45\right)±15}{2\times 3}$$

The opposite of -45 is 45.

$$x=\frac{45±15}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{45±15}{6}$$

Now solve the equation x=\frac{45±15}{6} when ± is plus. Add 45 to 15.

$$x=\frac{60}{6}$$

Divide 60 by 6.

$$x=10$$

Now solve the equation x=\frac{45±15}{6} when ± is minus. Subtract 15 from 45.

$$x=\frac{30}{6}$$

Divide 30 by 6.

$$x=5$$

The equation is now solved.

$$x=10 x=5$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,15 since division by zero is not defined. Multiply both sides of the equation by 20x\left(x-15\right), the least common multiple of x,15-x,20.

$$\left(20x-300\right)\times \frac{1}{2}-20x\times \frac{1}{2}=3x\left(x-15\right)$$

Use the distributive property to multiply 20x-300 by \frac{1}{2}.

$$10x-150-20x\times \frac{1}{2}=3x\left(x-15\right)$$

Multiply -20 and \frac{1}{2} to get -10.

$$10x-150-10x=3x\left(x-15\right)$$

Combine 10x and -10x to get 0.

$$-150=3x\left(x-15\right)$$

Use the distributive property to multiply 3x by x-15.

$$-150=3x^{2}-45x$$

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}-45x=-150$$

Divide both sides by 3.

$$\frac{3x^{2}-45x}{3}=-\frac{150}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{45}{3}\right)x=-\frac{150}{3}$$

Divide -45 by 3.

$$x^{2}-15x=-\frac{150}{3}$$

Divide -150 by 3.

$$x^{2}-15x=-50$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-50+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-50+\frac{225}{4}$$

Add -50 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=\frac{25}{4}$$

Factor x^{2}-15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5}{2} x-\frac{15}{2}=-\frac{5}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=10 x=5$$