Solve -4/3p+20/3=-p^2+20+3

$$- \frac { 4 } { 3 } p + \frac { 20 } { 3 } = - p ^ { 2 } + 20 + 3$$

Solve for p

$$p = \frac{\sqrt{151} + 2}{3} \approx 4.762735242$$

$$p=\frac{2-\sqrt{151}}{3}\approx -3.429401909$$

Steps Using the Quadratic Formula

Add 20 and 3 to get 23.

$$-\frac{4}{3}p+\frac{20}{3}=-p^{2}+23$$

Add p^{2} to both sides.

$$-\frac{4}{3}p+\frac{20}{3}+p^{2}=23$$

Subtract 23 from both sides.

$$-\frac{4}{3}p+\frac{20}{3}+p^{2}-23=0$$

Subtract 23 from \frac{20}{3} to get -\frac{49}{3}.

$$-\frac{4}{3}p-\frac{49}{3}+p^{2}=0$$

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.

$$p^{2}-\frac{4}{3}p-\frac{49}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{4}{3} for b, and -\frac{49}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\left(-\frac{4}{3}\right)^{2}-4\left(-\frac{49}{3}\right)}}{2}$$

Square -\frac{4}{3} by squaring both the numerator and the denominator of the fraction.

$$p=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}-4\left(-\frac{49}{3}\right)}}{2}$$

Multiply -4 times -\frac{49}{3}.

$$p=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{16}{9}+\frac{196}{3}}}{2}$$

Add \frac{16}{9} to \frac{196}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-\left(-\frac{4}{3}\right)±\sqrt{\frac{604}{9}}}{2}$$

Take the square root of \frac{604}{9}.

$$p=\frac{-\left(-\frac{4}{3}\right)±\frac{2\sqrt{151}}{3}}{2}$$

The opposite of -\frac{4}{3} is \frac{4}{3}.

$$p=\frac{\frac{4}{3}±\frac{2\sqrt{151}}{3}}{2}$$

Now solve the equation p=\frac{\frac{4}{3}±\frac{2\sqrt{151}}{3}}{2} when ± is plus. Add \frac{4}{3} to \frac{2\sqrt{151}}{3}.

$$p=\frac{2\sqrt{151}+4}{2\times 3}$$

Divide \frac{4+2\sqrt{151}}{3} by 2.

$$p=\frac{\sqrt{151}+2}{3}$$

Now solve the equation p=\frac{\frac{4}{3}±\frac{2\sqrt{151}}{3}}{2} when ± is minus. Subtract \frac{2\sqrt{151}}{3} from \frac{4}{3}.

$$p=\frac{4-2\sqrt{151}}{2\times 3}$$

Divide \frac{4-2\sqrt{151}}{3} by 2.

$$p=\frac{2-\sqrt{151}}{3}$$

The equation is now solved.

$$p=\frac{\sqrt{151}+2}{3} p=\frac{2-\sqrt{151}}{3}$$

Steps for Completing the Square

Add 20 and 3 to get 23.

$$-\frac{4}{3}p+\frac{20}{3}=-p^{2}+23$$

Add p^{2} to both sides.

$$-\frac{4}{3}p+\frac{20}{3}+p^{2}=23$$

Subtract \frac{20}{3} from both sides.

$$-\frac{4}{3}p+p^{2}=23-\frac{20}{3}$$

Subtract \frac{20}{3} from 23 to get \frac{49}{3}.

$$-\frac{4}{3}p+p^{2}=\frac{49}{3}$$

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.

$$p^{2}-\frac{4}{3}p=\frac{49}{3}$$

Divide -\frac{4}{3}, the coefficient of the x term, by 2 to get -\frac{2}{3}. Then add the square of -\frac{2}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{4}{3}p+\left(-\frac{2}{3}\right)^{2}=\frac{49}{3}+\left(-\frac{2}{3}\right)^{2}$$

Square -\frac{2}{3} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{4}{3}p+\frac{4}{9}=\frac{49}{3}+\frac{4}{9}$$

Add \frac{49}{3} to \frac{4}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-\frac{4}{3}p+\frac{4}{9}=\frac{151}{9}$$

Factor p^{2}-\frac{4}{3}p+\frac{4}{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}.

$$\left(p-\frac{2}{3}\right)^{2}=\frac{151}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{2}{3}\right)^{2}}=\sqrt{\frac{151}{9}}$$

Simplify.

$$p-\frac{2}{3}=\frac{\sqrt{151}}{3} p-\frac{2}{3}=-\frac{\sqrt{151}}{3}$$

Add \frac{2}{3} to both sides of the equation.

$$p=\frac{\sqrt{151}+2}{3} p=\frac{2-\sqrt{151}}{3}$$