Solve 12/p-24=3p-13/p-24-3/p

$$\frac { 12 } { p - 24 } = \frac { 3 p - 13 } { p - 24 } - \frac { 3 } { p }$$

Solve for p

$$p=\frac{14+2\sqrt{5}i}{3}\approx 4.666666667+1.490711985i$$

$$p=\frac{-2\sqrt{5}i+14}{3}\approx 4.666666667-1.490711985i$$

Steps Using the Quadratic Formula

Variable p cannot be equal to any of the values 0,24 since division by zero is not defined. Multiply both sides of the equation by p\left(p-24\right), the least common multiple of p-24,p.

$$p\times 12=p\left(3p-13\right)-\left(p-24\right)\times 3$$

Use the distributive property to multiply p by 3p-13.

$$p\times 12=3p^{2}-13p-\left(p-24\right)\times 3$$

Use the distributive property to multiply p-24 by 3.

$$p\times 12=3p^{2}-13p-\left(3p-72\right)$$

To find the opposite of 3p-72, find the opposite of each term.

$$p\times 12=3p^{2}-13p-3p+72$$

Combine -13p and -3p to get -16p.

$$p\times 12=3p^{2}-16p+72$$

Subtract 3p^{2} from both sides.

$$p\times 12-3p^{2}=-16p+72$$

Add 16p to both sides.

$$p\times 12-3p^{2}+16p=72$$

Combine p\times 12 and 16p to get 28p.

$$28p-3p^{2}=72$$

Subtract 72 from both sides.

$$28p-3p^{2}-72=0$$

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.

$$-3p^{2}+28p-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 28 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-28±\sqrt{28^{2}-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Square 28.

$$p=\frac{-28±\sqrt{784-4\left(-3\right)\left(-72\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$p=\frac{-28±\sqrt{784+12\left(-72\right)}}{2\left(-3\right)}$$

Multiply 12 times -72.

$$p=\frac{-28±\sqrt{784-864}}{2\left(-3\right)}$$

Add 784 to -864.

$$p=\frac{-28±\sqrt{-80}}{2\left(-3\right)}$$

Take the square root of -80.

$$p=\frac{-28±4\sqrt{5}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$p=\frac{-28±4\sqrt{5}i}{-6}$$

Now solve the equation p=\frac{-28±4\sqrt{5}i}{-6} when ± is plus. Add -28 to 4i\sqrt{5}.

$$p=\frac{-28+4\sqrt{5}i}{-6}$$

Divide -28+4i\sqrt{5} by -6.

$$p=\frac{-2\sqrt{5}i+14}{3}$$

Now solve the equation p=\frac{-28±4\sqrt{5}i}{-6} when ± is minus. Subtract 4i\sqrt{5} from -28.

$$p=\frac{-4\sqrt{5}i-28}{-6}$$

Divide -28-4i\sqrt{5} by -6.

$$p=\frac{14+2\sqrt{5}i}{3}$$

The equation is now solved.

$$p=\frac{-2\sqrt{5}i+14}{3} p=\frac{14+2\sqrt{5}i}{3}$$

Steps for Completing the Square

Variable p cannot be equal to any of the values 0,24 since division by zero is not defined. Multiply both sides of the equation by p\left(p-24\right), the least common multiple of p-24,p.

$$p\times 12=p\left(3p-13\right)-\left(p-24\right)\times 3$$

Use the distributive property to multiply p by 3p-13.

$$p\times 12=3p^{2}-13p-\left(p-24\right)\times 3$$

Use the distributive property to multiply p-24 by 3.

$$p\times 12=3p^{2}-13p-\left(3p-72\right)$$

To find the opposite of 3p-72, find the opposite of each term.

$$p\times 12=3p^{2}-13p-3p+72$$

Combine -13p and -3p to get -16p.

$$p\times 12=3p^{2}-16p+72$$

Subtract 3p^{2} from both sides.

$$p\times 12-3p^{2}=-16p+72$$

Add 16p to both sides.

$$p\times 12-3p^{2}+16p=72$$

Combine p\times 12 and 16p to get 28p.

$$28p-3p^{2}=72$$

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.

$$-3p^{2}+28p=72$$

Divide both sides by -3.

$$\frac{-3p^{2}+28p}{-3}=\frac{72}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$p^{2}+\frac{28}{-3}p=\frac{72}{-3}$$

Divide 28 by -3.

$$p^{2}-\frac{28}{3}p=\frac{72}{-3}$$

Divide 72 by -3.

$$p^{2}-\frac{28}{3}p=-24$$

Divide -\frac{28}{3}, the coefficient of the x term, by 2 to get -\frac{14}{3}. Then add the square of -\frac{14}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{28}{3}p+\left(-\frac{14}{3}\right)^{2}=-24+\left(-\frac{14}{3}\right)^{2}$$

Square -\frac{14}{3} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{28}{3}p+\frac{196}{9}=-24+\frac{196}{9}$$

Add -24 to \frac{196}{9}.

$$p^{2}-\frac{28}{3}p+\frac{196}{9}=-\frac{20}{9}$$

Factor p^{2}-\frac{28}{3}p+\frac{196}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{14}{3}\right)^{2}=-\frac{20}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{14}{3}\right)^{2}}=\sqrt{-\frac{20}{9}}$$

Simplify.

$$p-\frac{14}{3}=\frac{2\sqrt{5}i}{3} p-\frac{14}{3}=-\frac{2\sqrt{5}i}{3}$$

Add \frac{14}{3} to both sides of the equation.

$$p=\frac{14+2\sqrt{5}i}{3} p=\frac{-2\sqrt{5}i+14}{3}$$