Solve 4183.92+2*78x=6.5*10^4x^2

$$4183.92 + 2 \times 78 x = 6.5 \times 10 ^ { 4 } x ^ { 2 }$$

Solve for x

$$x=\frac{3\sqrt{44701}+3}{2500}\approx 0.254911332$$

$$x=\frac{3-3\sqrt{44701}}{2500}\approx -0.252511332$$

Steps Using the Quadratic Formula

Multiply 2 and 78 to get 156.

$$4183.92+156x=6.5\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$4183.92+156x=6.5\times 10000x^{2}$$

Multiply 6.5 and 10000 to get 65000.

$$4183.92+156x=65000x^{2}$$

Subtract 65000x^{2} from both sides.

$$4183.92+156x-65000x^{2}=0$$

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.

$$-65000x^{2}+156x+4183.92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -65000 for a, 156 for b, and 4183.92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-156±\sqrt{156^{2}-4\left(-65000\right)\times 4183.92}}{2\left(-65000\right)}$$

Square 156.

$$x=\frac{-156±\sqrt{24336-4\left(-65000\right)\times 4183.92}}{2\left(-65000\right)}$$

Multiply -4 times -65000.

$$x=\frac{-156±\sqrt{24336+260000\times 4183.92}}{2\left(-65000\right)}$$

Multiply 260000 times 4183.92.

$$x=\frac{-156±\sqrt{24336+1087819200}}{2\left(-65000\right)}$$

Add 24336 to 1087819200.

$$x=\frac{-156±\sqrt{1087843536}}{2\left(-65000\right)}$$

Take the square root of 1087843536.

$$x=\frac{-156±156\sqrt{44701}}{2\left(-65000\right)}$$

Multiply 2 times -65000.

$$x=\frac{-156±156\sqrt{44701}}{-130000}$$

Now solve the equation x=\frac{-156±156\sqrt{44701}}{-130000} when ± is plus. Add -156 to 156\sqrt{44701}.

$$x=\frac{156\sqrt{44701}-156}{-130000}$$

Divide -156+156\sqrt{44701} by -130000.

$$x=\frac{3-3\sqrt{44701}}{2500}$$

Now solve the equation x=\frac{-156±156\sqrt{44701}}{-130000} when ± is minus. Subtract 156\sqrt{44701} from -156.

$$x=\frac{-156\sqrt{44701}-156}{-130000}$$

Divide -156-156\sqrt{44701} by -130000.

$$x=\frac{3\sqrt{44701}+3}{2500}$$

The equation is now solved.

$$x=\frac{3-3\sqrt{44701}}{2500} x=\frac{3\sqrt{44701}+3}{2500}$$

Steps for Completing the Square

Multiply 2 and 78 to get 156.

$$4183.92+156x=6.5\times 10^{4}x^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$4183.92+156x=6.5\times 10000x^{2}$$

Multiply 6.5 and 10000 to get 65000.

$$4183.92+156x=65000x^{2}$$

Subtract 65000x^{2} from both sides.

$$4183.92+156x-65000x^{2}=0$$

Subtract 4183.92 from both sides. Anything subtracted from zero gives its negation.

$$156x-65000x^{2}=-4183.92$$

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.

$$-65000x^{2}+156x=-4183.92$$

Divide both sides by -65000.

$$\frac{-65000x^{2}+156x}{-65000}=-\frac{4183.92}{-65000}$$

Dividing by -65000 undoes the multiplication by -65000.

$$x^{2}+\frac{156}{-65000}x=-\frac{4183.92}{-65000}$$

Reduce the fraction \frac{156}{-65000} to lowest terms by extracting and canceling out 52.

$$x^{2}-\frac{3}{1250}x=-\frac{4183.92}{-65000}$$

Divide -4183.92 by -65000.

$$x^{2}-\frac{3}{1250}x=0.064368$$

Divide -\frac{3}{1250}, the coefficient of the x term, by 2 to get -\frac{3}{2500}. Then add the square of -\frac{3}{2500} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{1250}x+\left(-\frac{3}{2500}\right)^{2}=0.064368+\left(-\frac{3}{2500}\right)^{2}$$

Square -\frac{3}{2500} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{1250}x+\frac{9}{6250000}=0.064368+\frac{9}{6250000}$$

Add 0.064368 to \frac{9}{6250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{1250}x+\frac{9}{6250000}=\frac{402309}{6250000}$$

Factor x^{2}-\frac{3}{1250}x+\frac{9}{6250000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{2500}\right)^{2}=\frac{402309}{6250000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2500}\right)^{2}}=\sqrt{\frac{402309}{6250000}}$$

Simplify.

$$x-\frac{3}{2500}=\frac{3\sqrt{44701}}{2500} x-\frac{3}{2500}=-\frac{3\sqrt{44701}}{2500}$$

Add \frac{3}{2500} to both sides of the equation.

$$x=\frac{3\sqrt{44701}+3}{2500} x=\frac{3-3\sqrt{44701}}{2500}$$