Solve 1.044*10^-3*p=8.3145*298.15*left(1-186*10^-6*p+1.06*10^-8*p^2right)

$$1.044 \times { 10 }^{ -3 } \times p = 8.3145 \times 298.15 \times \left( 1-186 \times { 10 }^{ -6 } \times p+1.06 \times { 10 }^{ -8 } \times { p }^{ 2 } \right)$$

Solve for p

$$p=\frac{15404402685000+i\times 5000\sqrt{2088619926374770231}}{1751804177}\approx 8793.450139718+4124.906942198i$$

$$p=\frac{-i\times 5000\sqrt{2088619926374770231}[phone]}{1751804177}\approx 8793.450139718-4124.906942198i$$

Steps Using the Quadratic Formula

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$1.044\times \frac{1}{1000}p=8.3145\times 298.15\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 1.044 and \frac{1}{1000} to get \frac{261}{250000}.

$$\frac{261}{250000}p=8.3145\times 298.15\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 8.3145 and 298.15 to get 2478.968175.

$$\frac{261}{250000}p=2478.968175\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{261}{250000}p=2478.968175\left(1-186\times \frac{1}{1000000}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 186 and \frac{1}{1000000} to get \frac{93}{500000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+1.06\times 10^{-8}p^{2}\right)$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+1.06\times \frac{1}{100000000}p^{2}\right)$$

Multiply 1.06 and \frac{1}{100000000} to get \frac{53}{5000000000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+\frac{53}{5000000000}p^{2}\right)$$

Use the distributive property to multiply 2478.968175 by 1-\frac{93}{500000}p+\frac{53}{5000000000}p^{2}.

$$\frac{261}{250000}p=2478.968175-\frac{9221761611}{20000000000}p+\frac{5255412531}{200000000000000}p^{2}$$

Subtract 2478.968175 from both sides.

$$\frac{261}{250000}p-2478.968175=-\frac{9221761611}{20000000000}p+\frac{5255412531}{200000000000000}p^{2}$$

Add \frac{9221761611}{20000000000}p to both sides.

$$\frac{261}{250000}p-2478.968175+\frac{9221761611}{20000000000}p=\frac{5255412531}{200000000000000}p^{2}$$

Combine \frac{261}{250000}p and \frac{9221761611}{20000000000}p to get \frac{9242641611}{20000000000}p.

$$\frac{9242641611}{20000000000}p-2478.968175=\frac{5255412531}{200000000000000}p^{2}$$

Subtract \frac{5255412531}{200000000000000}p^{2} from both sides.

$$\frac{9242641611}{20000000000}p-2478.968175-\frac{5255412531}{200000000000000}p^{2}=0$$

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.

$$-\frac{5255412531}{200000000000000}p^{2}+\frac{9242641611}{20000000000}p-2478.968175=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{5255412531}{200000000000000} for a, \frac{9242641611}{20000000000} for b, and -2478.968175 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\frac{9242641611}{20000000000}±\sqrt{\left(\frac{9242641611}{20000000000}\right)^{2}-4\left(-\frac{5255412531}{200000000000000}\right)\left(-2478.968175\right)}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Square \frac{9242641611}{20000000000} by squaring both the numerator and the denominator of the fraction.

$$p=\frac{-\frac{9242641611}{20000000000}±\sqrt{\frac{85426423949388675321}{400000000000000000000}-4\left(-\frac{5255412531}{200000000000000}\right)\left(-2478.968175\right)}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Multiply -4 times -\frac{5255412531}{200000000000000}.

$$p=\frac{-\frac{9242641611}{20000000000}±\sqrt{\frac{85426423949388675321}{400000000000000000000}+\frac{5255412531}{50000000000000}\left(-2478.968175\right)}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Multiply \frac{5255412531}{50000000000000} times -2478.968175 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-\frac{9242641611}{20000000000}±\sqrt{\frac{85426423949388675321}{400000000000000000000}-\frac{521120016433808037}{2000000000000000000}}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Add \frac{85426423949388675321}{400000000000000000000} to -\frac{521120016433808037}{2000000000000000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p=\frac{-\frac{9242641611}{20000000000}±\sqrt{-\frac{18797579337372932079}{400000000000000000000}}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Take the square root of -\frac{18797579337372932079}{400000000000000000000}.

$$p=\frac{-\frac{9242641611}{20000000000}±\frac{3\sqrt{2088619926374770231}i}{20000000000}}{2\left(-\frac{5255412531}{200000000000000}\right)}$$

Multiply 2 times -\frac{5255412531}{200000000000000}.

$$p=\frac{-\frac{9242641611}{20000000000}±\frac{3\sqrt{2088619926374770231}i}{20000000000}}{-\frac{5255412531}{100000000000000}}$$

Now solve the equation p=\frac{-\frac{9242641611}{20000000000}±\frac{3\sqrt{2088619926374770231}i}{20000000000}}{-\frac{5255412531}{100000000000000}} when ± is plus. Add -\frac{9242641611}{20000000000} to \frac{3i\sqrt{2088619926374770231}}{20000000000}.

$$p=\frac{-9242641611+3\sqrt{2088619926374770231}i}{-\frac{5255412531}{100000000000000}\times 20000000000}$$

Divide \frac{-9242641611+3i\sqrt{2088619926374770231}}{20000000000} by -\frac{5255412531}{100000000000000} by multiplying \frac{-9242641611+3i\sqrt{2088619926374770231}}{20000000000} by the reciprocal of -\frac{5255412531}{100000000000000}.

$$p=\frac{-5000\sqrt{2088619926374770231}i+15404402685000}{1751804177}$$

Now solve the equation p=\frac{-\frac{9242641611}{20000000000}±\frac{3\sqrt{2088619926374770231}i}{20000000000}}{-\frac{5255412531}{100000000000000}} when ± is minus. Subtract \frac{3i\sqrt{2088619926374770231}}{20000000000} from -\frac{9242641611}{20000000000}.

$$p=\frac{-3\sqrt{2088619926374770231}i-9242641611}{-\frac{5255412531}{100000000000000}\times 20000000000}$$

Divide \frac{-9242641611-3i\sqrt{2088619926374770231}}{20000000000} by -\frac{5255412531}{100000000000000} by multiplying \frac{-9242641611-3i\sqrt{2088619926374770231}}{20000000000} by the reciprocal of -\frac{5255412531}{100000000000000}.

$$p=\frac{15404402685000+5000\sqrt{2088619926374770231}i}{1751804177}$$

The equation is now solved.

$$p=\frac{-5000\sqrt{2088619926374770231}i+15404402685000}{1751804177} p=\frac{15404402685000+5000\sqrt{2088619926374770231}i}{1751804177}$$

Steps for Completing the Square

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$1.044\times \frac{1}{1000}p=8.3145\times 298.15\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 1.044 and \frac{1}{1000} to get \frac{261}{250000}.

$$\frac{261}{250000}p=8.3145\times 298.15\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 8.3145 and 298.15 to get 2478.968175.

$$\frac{261}{250000}p=2478.968175\left(1-186\times 10^{-6}p+1.06\times 10^{-8}p^{2}\right)$$

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\frac{261}{250000}p=2478.968175\left(1-186\times \frac{1}{1000000}p+1.06\times 10^{-8}p^{2}\right)$$

Multiply 186 and \frac{1}{1000000} to get \frac{93}{500000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+1.06\times 10^{-8}p^{2}\right)$$

Calculate 10 to the power of -8 and get \frac{1}{100000000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+1.06\times \frac{1}{100000000}p^{2}\right)$$

Multiply 1.06 and \frac{1}{100000000} to get \frac{53}{5000000000}.

$$\frac{261}{250000}p=2478.968175\left(1-\frac{93}{500000}p+\frac{53}{5000000000}p^{2}\right)$$

Use the distributive property to multiply 2478.968175 by 1-\frac{93}{500000}p+\frac{53}{5000000000}p^{2}.

$$\frac{261}{250000}p=2478.968175-\frac{9221761611}{20000000000}p+\frac{5255412531}{200000000000000}p^{2}$$

Add \frac{9221761611}{20000000000}p to both sides.

$$\frac{261}{250000}p+\frac{9221761611}{20000000000}p=2478.968175+\frac{5255412531}{200000000000000}p^{2}$$

Combine \frac{261}{250000}p and \frac{9221761611}{20000000000}p to get \frac{9242641611}{20000000000}p.

$$\frac{9242641611}{20000000000}p=2478.968175+\frac{5255412531}{200000000000000}p^{2}$$

Subtract \frac{5255412531}{200000000000000}p^{2} from both sides.

$$\frac{9242641611}{20000000000}p-\frac{5255412531}{200000000000000}p^{2}=2478.968175$$

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.

$$-\frac{5255412531}{200000000000000}p^{2}+\frac{9242641611}{20000000000}p=2478.968175$$

Divide both sides of the equation by -\frac{5255412531}{200000000000000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{5255412531}{200000000000000}p^{2}+\frac{9242641611}{20000000000}p}{-\frac{5255412531}{200000000000000}}=\frac{2478.968175}{-\frac{5255412531}{200000000000000}}$$

Dividing by -\frac{5255412531}{200000000000000} undoes the multiplication by -\frac{5255412531}{200000000000000}.

$$p^{2}+\frac{\frac{9242641611}{20000000000}}{-\frac{5255412531}{200000000000000}}p=\frac{2478.968175}{-\frac{5255412531}{200000000000000}}$$

Divide \frac{9242641611}{20000000000} by -\frac{5255412531}{200000000000000} by multiplying \frac{9242641611}{20000000000} by the reciprocal of -\frac{5255412531}{200000000000000}.

$$p^{2}-\frac{30808805370000}{1751804177}p=\frac{2478.968175}{-\frac{5255412531}{200000000000000}}$$

Divide 2478.968175 by -\frac{5255412531}{200000000000000} by multiplying 2478.968175 by the reciprocal of -\frac{5255412531}{200000000000000}.

$$p^{2}-\frac{30808805370000}{1751804177}p=-\frac{5000000000}{53}$$

Divide -\frac{30808805370000}{1751804177}, the coefficient of the x term, by 2 to get -\frac{15404402685000}{1751804177}. Then add the square of -\frac{15404402685000}{1751804177} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$p^{2}-\frac{30808805370000}{1751804177}p+\left(-\frac{15404402685000}{1751804177}\right)^{2}=-\frac{5000000000}{53}+\left(-\frac{15404402685000}{1751804177}\right)^{2}$$

Square -\frac{15404402685000}{1751804177} by squaring both the numerator and the denominator of the fraction.

$$p^{2}-\frac{30808805370000}{1751804177}p+\frac{237295622081635209225000000}{3068817874554647329}=-\frac{5000000000}{53}+\frac{237295622081635209225000000}{3068817874554647329}$$

Add -\frac{5000000000}{53} to \frac{237295622081635209225000000}{3068817874554647329} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$p^{2}-\frac{30808805370000}{1751804177}p+\frac{237295622081635209225000000}{3068817874554647329}=-\frac{52215498159369255775000000}{3068817874554647329}$$

Factor p^{2}-\frac{30808805370000}{1751804177}p+\frac{237295622081635209225000000}{3068817874554647329}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(p-\frac{15404402685000}{1751804177}\right)^{2}=-\frac{52215498159369255775000000}{3068817874554647329}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(p-\frac{15404402685000}{1751804177}\right)^{2}}=\sqrt{-\frac{52215498159369255775000000}{3068817874554647329}}$$

Simplify.

$$p-\frac{15404402685000}{1751804177}=\frac{5000\sqrt{2088619926374770231}i}{1751804177} p-\frac{15404402685000}{1751804177}=-\frac{5000\sqrt{2088619926374770231}i}{1751804177}$$

Add \frac{15404402685000}{1751804177} to both sides of the equation.

$$p=\frac{15404402685000+5000\sqrt{2088619926374770231}i}{1751804177} p=\frac{-5000\sqrt{2088619926374770231}i+15404402685000}{1751804177}$$