Solve p^2-2p-3/p-5=4/3

$$\frac { p ^ { 2 } - 2 p - 3 } { p - 5 } = \frac { 4 } { 3 }$$

Solve for p

$$p=\frac{5+2\sqrt{2}i}{3}\approx 1.666666667+0.942809042i$$

$$p=\frac{-2\sqrt{2}i+5}{3}\approx 1.666666667-0.942809042i$$

Steps Using the Quadratic Formula

Variable p cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 3\left(p-5\right), the least common multiple of p-5,3.

$$3\left(p^{2}-2p-3\right)=4\left(p-5\right)$$

Use the distributive property to multiply 3 by p^{2}-2p-3.

$$3p^{2}-6p-9=4\left(p-5\right)$$

Use the distributive property to multiply 4 by p-5.

$$3p^{2}-6p-9=4p-20$$

Subtract 4p from both sides.

$$3p^{2}-6p-9-4p=-20$$

Combine -6p and -4p to get -10p.

$$3p^{2}-10p-9=-20$$

Add 20 to both sides.

$$3p^{2}-10p-9+20=0$$

Add -9 and 20 to get 11.

$$3p^{2}-10p+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -10 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\times 11}}{2\times 3}$$

Square -10.

$$p=\frac{-\left(-10\right)±\sqrt{100-4\times 3\times 11}}{2\times 3}$$

Multiply -4 times 3.

$$p=\frac{-\left(-10\right)±\sqrt{100-12\times 11}}{2\times 3}$$

Multiply -12 times 11.

$$p=\frac{-\left(-10\right)±\sqrt{100-132}}{2\times 3}$$

Add 100 to -132.

$$p=\frac{-\left(-10\right)±\sqrt{-32}}{2\times 3}$$

Take the square root of -32.

$$p=\frac{-\left(-10\right)±4\sqrt{2}i}{2\times 3}$$

The opposite of -10 is 10.

$$p=\frac{10±4\sqrt{2}i}{2\times 3}$$

Multiply 2 times 3.

$$p=\frac{10±4\sqrt{2}i}{6}$$

Now solve the equation p=\frac{10±4\sqrt{2}i}{6} when ± is plus. Add 10 to 4i\sqrt{2}.

$$p=\frac{10+4\sqrt{2}i}{6}$$

Divide 10+4i\sqrt{2} by 6.

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

Now solve the equation p=\frac{10±4\sqrt{2}i}{6} when ± is minus. Subtract 4i\sqrt{2} from 10.

$$p=\frac{-4\sqrt{2}i+10}{6}$$

Divide 10-4i\sqrt{2} by 6.

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

The equation is now solved.

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

Steps for Completing the Square

Variable p cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by 3\left(p-5\right), the least common multiple of p-5,3.

$$3\left(p^{2}-2p-3\right)=4\left(p-5\right)$$

Use the distributive property to multiply 3 by p^{2}-2p-3.

$$3p^{2}-6p-9=4\left(p-5\right)$$

Use the distributive property to multiply 4 by p-5.

$$3p^{2}-6p-9=4p-20$$

Subtract 4p from both sides.

$$3p^{2}-6p-9-4p=-20$$

Combine -6p and -4p to get -10p.

$$3p^{2}-10p-9=-20$$

Add 9 to both sides.

$$3p^{2}-10p=-20+9$$

Add -20 and 9 to get -11.

$$3p^{2}-10p=-11$$

Divide both sides by 3.

$$\frac{3p^{2}-10p}{3}=-\frac{11}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$p^{2}-\frac{10}{3}p=-\frac{11}{3}$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{10}{3}p+\left(-\frac{5}{3}\right)^{2}=-\frac{11}{3}+\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{10}{3}p+\frac{25}{9}=-\frac{11}{3}+\frac{25}{9}$$

Add -\frac{11}{3} to \frac{25}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-\frac{10}{3}p+\frac{25}{9}=-\frac{8}{9}$$

Factor p^{2}-\frac{10}{3}p+\frac{25}{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}.

$$\left(p-\frac{5}{3}\right)^{2}=-\frac{8}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{5}{3}\right)^{2}}=\sqrt{-\frac{8}{9}}$$

Simplify.

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

Add \frac{5}{3} to both sides of the equation.

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