Solve 29500x^2-764.4x=4024.8

$$29500 x ^ { 2 } - 764.4 x = 4024.8$$

Solve for x

$$x=\frac{9\sqrt{36690641}+1911}{147500}\approx 0.382552678$$

$$x=\frac{1911-9\sqrt{36690641}}{147500}\approx -0.356640814$$

Steps Using the Quadratic Formula

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.

$$29500x^{2}-764.4x=4024.8$$

Subtract 4024.8 from both sides of the equation.

$$29500x^{2}-764.4x-4024.8=4024.8-4024.8$$

Subtracting 4024.8 from itself leaves 0.

$$29500x^{2}-764.4x-4024.8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 29500 for a, -764.4 for b, and -4024.8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-764.4\right)±\sqrt{\left(-764.4\right)^{2}-4\times 29500\left(-4024.8\right)}}{2\times 29500}$$

Square -764.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-764.4\right)±\sqrt{584307.36-4\times 29500\left(-4024.8\right)}}{2\times 29500}$$

Multiply -4 times 29500.

$$x=\frac{-\left(-764.4\right)±\sqrt{584307.36-118000\left(-4024.8\right)}}{2\times 29500}$$

Multiply -118000 times -4024.8.

$$x=\frac{-\left(-764.4\right)±\sqrt{584307.36+474926400}}{2\times 29500}$$

Add 584307.36 to 474926400.

$$x=\frac{-\left(-764.4\right)±\sqrt{475510707.36}}{2\times 29500}$$

Take the square root of 475510707.36.

$$x=\frac{-\left(-764.4\right)±\frac{18\sqrt{36690641}}{5}}{2\times 29500}$$

The opposite of -764.4 is 764.4.

$$x=\frac{764.4±\frac{18\sqrt{36690641}}{5}}{2\times 29500}$$

Multiply 2 times 29500.

$$x=\frac{764.4±\frac{18\sqrt{36690641}}{5}}{59000}$$

Now solve the equation x=\frac{764.4±\frac{18\sqrt{36690641}}{5}}{59000} when ± is plus. Add 764.4 to \frac{18\sqrt{36690641}}{5}.

$$x=\frac{18\sqrt{36690641}+3822}{5\times 59000}$$

Divide \frac{3822+18\sqrt{36690641}}{5} by 59000.

$$x=\frac{9\sqrt{36690641}+1911}{147500}$$

Now solve the equation x=\frac{764.4±\frac{18\sqrt{36690641}}{5}}{59000} when ± is minus. Subtract \frac{18\sqrt{36690641}}{5} from 764.4.

$$x=\frac{3822-18\sqrt{36690641}}{5\times 59000}$$

Divide \frac{3822-18\sqrt{36690641}}{5} by 59000.

$$x=\frac{1911-9\sqrt{36690641}}{147500}$$

The equation is now solved.

$$x=\frac{9\sqrt{36690641}+1911}{147500} x=\frac{1911-9\sqrt{36690641}}{147500}$$

Steps for Completing the Square

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.

$$29500x^{2}-764.4x=4024.8$$

Divide both sides by 29500.

$$\frac{29500x^{2}-764.4x}{29500}=\frac{4024.8}{29500}$$

Dividing by 29500 undoes the multiplication by 29500.

$$x^{2}+\left(-\frac{764.4}{29500}\right)x=\frac{4024.8}{29500}$$

Divide -764.4 by 29500.

$$x^{2}-\frac{1911}{73750}x=\frac{4024.8}{29500}$$

Divide 4024.8 by 29500.

$$x^{2}-\frac{1911}{73750}x=\frac{5031}{36875}$$

Divide -\frac{1911}{73750}, the coefficient of the x term, by 2 to get -\frac{1911}{147500}. Then add the square of -\frac{1911}{147500} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1911}{73750}x+\left(-\frac{1911}{147500}\right)^{2}=\frac{5031}{36875}+\left(-\frac{1911}{147500}\right)^{2}$$

Square -\frac{1911}{147500} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1911}{73750}x+\frac{3651921}{21756250000}=\frac{5031}{36875}+\frac{3651921}{21756250000}$$

Add \frac{5031}{36875} to \frac{3651921}{21756250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1911}{73750}x+\frac{3651921}{21756250000}=\frac{2971941921}{21756250000}$$

Factor x^{2}-\frac{1911}{73750}x+\frac{3651921}{21756250000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1911}{147500}\right)^{2}=\frac{2971941921}{21756250000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1911}{147500}\right)^{2}}=\sqrt{\frac{2971941921}{21756250000}}$$

Simplify.

$$x-\frac{1911}{147500}=\frac{9\sqrt{36690641}}{147500} x-\frac{1911}{147500}=-\frac{9\sqrt{36690641}}{147500}$$

Add \frac{1911}{147500} to both sides of the equation.

$$x=\frac{9\sqrt{36690641}+1911}{147500} x=\frac{1911-9\sqrt{36690641}}{147500}$$