Solve 958.46109p=2478.96818+0.000026277p^2

$$958.46109 p = 2478.96818 + 0.000026277 p ^ { 2 }$$

Solve for p

$$p=\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}\approx 36475283.405146755$$

$$p=-\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}\approx 2.586404793$$

Steps Using the Quadratic Formula

Subtract 2478.96818 from both sides.

$$958.46109p-2478.96818=0.000026277p^{2}$$

Subtract 0.000026277p^{2} from both sides.

$$958.46109p-2478.96818-0.000026277p^{2}=0$$

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.

$$-0.000026277p^{2}+958.46109p-2478.96818=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.000026277 for a, 958.46109 for b, and -2478.96818 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-958.46109±\sqrt{958.46109^{2}-4\left(-0.000026277\right)\left(-2478.96818\right)}}{2\left(-0.000026277\right)}$$

Square 958.46109 by squaring both the numerator and the denominator of the fraction.

$$p=\frac{-958.46109±\sqrt{918647.6610439881-4\left(-0.000026277\right)\left(-2478.96818\right)}}{2\left(-0.000026277\right)}$$

Multiply -4 times -0.000026277.

$$p=\frac{-958.46109±\sqrt{918647.6610439881+0.000105108\left(-2478.96818\right)}}{2\left(-0.000026277\right)}$$

Multiply 0.000105108 times -2478.96818 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-958.46109±\sqrt{918647.6610439881-0.26055938746344}}{2\left(-0.000026277\right)}$$

Add 918647.6610439881 to -0.26055938746344 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-958.46109±\sqrt{918647.40048460063656}}{2\left(-0.000026277\right)}$$

Take the square root of 918647.40048460063656.

$$p=\frac{-958.46109±\frac{\sqrt{22966185012115015914}}{5000000}}{2\left(-0.000026277\right)}$$

Multiply 2 times -0.000026277.

$$p=\frac{-958.46109±\frac{\sqrt{22966185012115015914}}{5000000}}{-0.000052554}$$

Now solve the equation p=\frac{-958.46109±\frac{\sqrt{22966185012115015914}}{5000000}}{-0.000052554} when ± is plus. Add -958.46109 to \frac{\sqrt{22966185012115015914}}{5000000}.

$$p=\frac{\frac{\sqrt{22966185012115015914}}{5000000}-\frac{95846109}{100000}}{-0.000052554}$$

Divide -\frac{95846109}{100000}+\frac{\sqrt{22966185012115015914}}{5000000} by -0.000052554 by multiplying -\frac{95846109}{100000}+\frac{\sqrt{22966185012115015914}}{5000000} by the reciprocal of -0.000052554.

$$p=-\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}$$

Now solve the equation p=\frac{-958.46109±\frac{\sqrt{22966185012115015914}}{5000000}}{-0.000052554} when ± is minus. Subtract \frac{\sqrt{22966185012115015914}}{5000000} from -958.46109.

$$p=\frac{-\frac{\sqrt{22966185012115015914}}{5000000}-\frac{95846109}{100000}}{-0.000052554}$$

Divide -\frac{95846109}{100000}-\frac{\sqrt{22966185012115015914}}{5000000} by -0.000052554 by multiplying -\frac{95846109}{100000}-\frac{\sqrt{22966185012115015914}}{5000000} by the reciprocal of -0.000052554.

$$p=\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}$$

The equation is now solved.

$$p=-\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759} p=\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}$$

Steps for Completing the Square

Subtract 0.000026277p^{2} from both sides.

$$958.46109p-0.000026277p^{2}=2478.96818$$

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.

$$-0.000026277p^{2}+958.46109p=2478.96818$$

Divide both sides of the equation by -0.000026277, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.000026277p^{2}+958.46109p}{-0.000026277}=\frac{2478.96818}{-0.000026277}$$

Dividing by -0.000026277 undoes the multiplication by -0.000026277.

$$p^{2}+\frac{958.46109}{-0.000026277}p=\frac{2478.96818}{-0.000026277}$$

Divide 958.46109 by -0.000026277 by multiplying 958.46109 by the reciprocal of -0.000026277.

$$p^{2}-\frac{319487030000}{8759}p=\frac{2478.96818}{-0.000026277}$$

Divide 2478.96818 by -0.000026277 by multiplying 2478.96818 by the reciprocal of -0.000026277.

$$p^{2}-\frac{319487030000}{8759}p=-\frac{2478968180000}{26277}$$

Divide -\frac{319487030000}{8759}, the coefficient of the x term, by 2 to get -\frac{159743515000}{8759}. Then add the square of -\frac{159743515000}{8759} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{319487030000}{8759}p+\left(-\frac{159743515000}{8759}\right)^{2}=-\frac{2478968180000}{26277}+\left(-\frac{159743515000}{8759}\right)^{2}$$

Square -\frac{159743515000}{8759} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{319487030000}{8759}p+\frac{25517990584555225000000}{76720081}=-\frac{2478968180000}{26277}+\frac{25517990584555225000000}{76720081}$$

Add -\frac{2478968180000}{26277} to \frac{25517990584555225000000}{76720081} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-\frac{319487030000}{8759}p+\frac{25517990584555225000000}{76720081}=\frac{76553950040383386380000}{230160243}$$

Factor p^{2}-\frac{319487030000}{8759}p+\frac{25517990584555225000000}{76720081}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{159743515000}{8759}\right)^{2}=\frac{76553950040383386380000}{230160243}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{159743515000}{8759}\right)^{2}}=\sqrt{\frac{76553950040383386380000}{230160243}}$$

Simplify.

$$p-\frac{159743515000}{8759}=\frac{100\sqrt{22966185012115015914}}{26277} p-\frac{159743515000}{8759}=-\frac{100\sqrt{22966185012115015914}}{26277}$$

Add \frac{159743515000}{8759} to both sides of the equation.

$$p=\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759} p=-\frac{100\sqrt{22966185012115015914}}{26277}+\frac{159743515000}{8759}$$