Solve 12/5(x^2+41)=365(10x+19)

$$\frac{ 12 }{ 5 } ( { x }^{ 2 } +41)=365(10x+19)$$

Solve for x

$$x = \frac{\sqrt{83675821} + 9125}{12} \approx 1522.704073278$$

$$x=\frac{9125-\sqrt{83675821}}{12}\approx -1.870739944$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{12}{5} by x^{2}+41.

$$\frac{12}{5}x^{2}+\frac{492}{5}=365\left(10x+19\right)$$

Use the distributive property to multiply 365 by 10x+19.

$$\frac{12}{5}x^{2}+\frac{492}{5}=3650x+6935$$

Subtract 3650x from both sides.

$$\frac{12}{5}x^{2}+\frac{492}{5}-3650x=6935$$

Subtract 6935 from both sides.

$$\frac{12}{5}x^{2}+\frac{492}{5}-3650x-6935=0$$

Subtract 6935 from \frac{492}{5} to get -\frac{34183}{5}.

$$\frac{12}{5}x^{2}-\frac{34183}{5}-3650x=0$$

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.

$$\frac{12}{5}x^{2}-3650x-\frac{34183}{5}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{12}{5} for a, -3650 for b, and -\frac{34183}{5} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3650\right)±\sqrt{\left(-3650\right)^{2}-4\times \frac{12}{5}\left(-\frac{34183}{5}\right)}}{2\times \frac{12}{5}}$$

Square -3650.

$$x=\frac{-\left(-3650\right)±\sqrt{13322500-4\times \frac{12}{5}\left(-\frac{34183}{5}\right)}}{2\times \frac{12}{5}}$$

Multiply -4 times \frac{12}{5}.

$$x=\frac{-\left(-3650\right)±\sqrt{13322500-\frac{48}{5}\left(-\frac{34183}{5}\right)}}{2\times \frac{12}{5}}$$

Multiply -\frac{48}{5} times -\frac{34183}{5} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-3650\right)±\sqrt{13322500+\frac{1640784}{25}}}{2\times \frac{12}{5}}$$

Add 13322500 to \frac{1640784}{25}.

$$x=\frac{-\left(-3650\right)±\sqrt{\frac{334703284}{25}}}{2\times \frac{12}{5}}$$

Take the square root of \frac{334703284}{25}.

$$x=\frac{-\left(-3650\right)±\frac{2\sqrt{83675821}}{5}}{2\times \frac{12}{5}}$$

The opposite of -3650 is 3650.

$$x=\frac{3650±\frac{2\sqrt{83675821}}{5}}{2\times \frac{12}{5}}$$

Multiply 2 times \frac{12}{5}.

$$x=\frac{3650±\frac{2\sqrt{83675821}}{5}}{\frac{24}{5}}$$

Now solve the equation x=\frac{3650±\frac{2\sqrt{83675821}}{5}}{\frac{24}{5}} when ± is plus. Add 3650 to \frac{2\sqrt{83675821}}{5}.

$$x=\frac{\frac{2\sqrt{83675821}}{5}+3650}{\frac{24}{5}}$$

Divide 3650+\frac{2\sqrt{83675821}}{5} by \frac{24}{5} by multiplying 3650+\frac{2\sqrt{83675821}}{5} by the reciprocal of \frac{24}{5}.

$$x=\frac{\sqrt{83675821}+9125}{12}$$

Now solve the equation x=\frac{3650±\frac{2\sqrt{83675821}}{5}}{\frac{24}{5}} when ± is minus. Subtract \frac{2\sqrt{83675821}}{5} from 3650.

$$x=\frac{-\frac{2\sqrt{83675821}}{5}+3650}{\frac{24}{5}}$$

Divide 3650-\frac{2\sqrt{83675821}}{5} by \frac{24}{5} by multiplying 3650-\frac{2\sqrt{83675821}}{5} by the reciprocal of \frac{24}{5}.

$$x=\frac{9125-\sqrt{83675821}}{12}$$

The equation is now solved.

$$x=\frac{\sqrt{83675821}+9125}{12} x=\frac{9125-\sqrt{83675821}}{12}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{12}{5} by x^{2}+41.

$$\frac{12}{5}x^{2}+\frac{492}{5}=365\left(10x+19\right)$$

Use the distributive property to multiply 365 by 10x+19.

$$\frac{12}{5}x^{2}+\frac{492}{5}=3650x+6935$$

Subtract 3650x from both sides.

$$\frac{12}{5}x^{2}+\frac{492}{5}-3650x=6935$$

Subtract \frac{492}{5} from both sides.

$$\frac{12}{5}x^{2}-3650x=6935-\frac{492}{5}$$

Subtract \frac{492}{5} from 6935 to get \frac{34183}{5}.

$$\frac{12}{5}x^{2}-3650x=\frac{34183}{5}$$

Divide both sides of the equation by \frac{12}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{12}{5}x^{2}-3650x}{\frac{12}{5}}=\frac{\frac{34183}{5}}{\frac{12}{5}}$$

Dividing by \frac{12}{5} undoes the multiplication by \frac{12}{5}.

$$x^{2}+\left(-\frac{3650}{\frac{12}{5}}\right)x=\frac{\frac{34183}{5}}{\frac{12}{5}}$$

Divide -3650 by \frac{12}{5} by multiplying -3650 by the reciprocal of \frac{12}{5}.

$$x^{2}-\frac{9125}{6}x=\frac{\frac{34183}{5}}{\frac{12}{5}}$$

Divide \frac{34183}{5} by \frac{12}{5} by multiplying \frac{34183}{5} by the reciprocal of \frac{12}{5}.

$$x^{2}-\frac{9125}{6}x=\frac{34183}{12}$$

Divide -\frac{9125}{6}, the coefficient of the x term, by 2 to get -\frac{9125}{12}. Then add the square of -\frac{9125}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9125}{6}x+\left(-\frac{9125}{12}\right)^{2}=\frac{34183}{12}+\left(-\frac{9125}{12}\right)^{2}$$

Square -\frac{9125}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9125}{6}x+\frac{83265625}{144}=\frac{34183}{12}+\frac{83265625}{144}$$

Add \frac{34183}{12} to \frac{83265625}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9125}{6}x+\frac{83265625}{144}=\frac{83675821}{144}$$

Factor x^{2}-\frac{9125}{6}x+\frac{83265625}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9125}{12}\right)^{2}=\frac{83675821}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9125}{12}\right)^{2}}=\sqrt{\frac{83675821}{144}}$$

Simplify.

$$x-\frac{9125}{12}=\frac{\sqrt{83675821}}{12} x-\frac{9125}{12}=-\frac{\sqrt{83675821}}{12}$$

Add \frac{9125}{12} to both sides of the equation.

$$x=\frac{\sqrt{83675821}+9125}{12} x=\frac{9125-\sqrt{83675821}}{12}$$