Solve 10.4x-80.4=0.179x^2

$$10.4x-80.4=0.179 { x }^{ 2 }$$

Solve for x

$$x = \frac{20 \sqrt{31621} + 5200}{179} \approx 48.918764877$$

$$x = \frac{5200 - 20 \sqrt{31621}}{179} \approx 9.181793782$$

Steps Using the Quadratic Formula

Subtract 0.179x^{2} from both sides.

$$10.4x-80.4-0.179x^{2}=0$$

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.

$$-0.179x^{2}+10.4x-80.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.179 for a, 10.4 for b, and -80.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10.4±\sqrt{10.4^{2}-4\left(-0.179\right)\left(-80.4\right)}}{2\left(-0.179\right)}$$

Square 10.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-10.4±\sqrt{108.16-4\left(-0.179\right)\left(-80.4\right)}}{2\left(-0.179\right)}$$

Multiply -4 times -0.179.

$$x=\frac{-10.4±\sqrt{108.16+0.716\left(-80.4\right)}}{2\left(-0.179\right)}$$

Multiply 0.716 times -80.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-10.4±\sqrt{108.16-57.5664}}{2\left(-0.179\right)}$$

Add 108.16 to -57.5664 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-10.4±\sqrt{50.5936}}{2\left(-0.179\right)}$$

Take the square root of 50.5936.

$$x=\frac{-10.4±\frac{\sqrt{31621}}{25}}{2\left(-0.179\right)}$$

Multiply 2 times -0.179.

$$x=\frac{-10.4±\frac{\sqrt{31621}}{25}}{-0.358}$$

Now solve the equation x=\frac{-10.4±\frac{\sqrt{31621}}{25}}{-0.358} when ± is plus. Add -10.4 to \frac{\sqrt{31621}}{25}.

$$x=\frac{\frac{\sqrt{31621}}{25}-\frac{52}{5}}{-0.358}$$

Divide -\frac{52}{5}+\frac{\sqrt{31621}}{25} by -0.358 by multiplying -\frac{52}{5}+\frac{\sqrt{31621}}{25} by the reciprocal of -0.358.

$$x=\frac{5200-20\sqrt{31621}}{179}$$

Now solve the equation x=\frac{-10.4±\frac{\sqrt{31621}}{25}}{-0.358} when ± is minus. Subtract \frac{\sqrt{31621}}{25} from -10.4.

$$x=\frac{-\frac{\sqrt{31621}}{25}-\frac{52}{5}}{-0.358}$$

Divide -\frac{52}{5}-\frac{\sqrt{31621}}{25} by -0.358 by multiplying -\frac{52}{5}-\frac{\sqrt{31621}}{25} by the reciprocal of -0.358.

$$x=\frac{20\sqrt{31621}+5200}{179}$$

The equation is now solved.

$$x=\frac{5200-20\sqrt{31621}}{179} x=\frac{20\sqrt{31621}+5200}{179}$$

Steps for Completing the Square

Subtract 0.179x^{2} from both sides.

$$10.4x-80.4-0.179x^{2}=0$$

Add 80.4 to both sides. Anything plus zero gives itself.

$$10.4x-0.179x^{2}=80.4$$

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.

$$-0.179x^{2}+10.4x=80.4$$

Divide both sides of the equation by -0.179, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.179x^{2}+10.4x}{-0.179}=\frac{80.4}{-0.179}$$

Dividing by -0.179 undoes the multiplication by -0.179.

$$x^{2}+\frac{10.4}{-0.179}x=\frac{80.4}{-0.179}$$

Divide 10.4 by -0.179 by multiplying 10.4 by the reciprocal of -0.179.

$$x^{2}-\frac{10400}{179}x=\frac{80.4}{-0.179}$$

Divide 80.4 by -0.179 by multiplying 80.4 by the reciprocal of -0.179.

$$x^{2}-\frac{10400}{179}x=-\frac{80400}{179}$$

Divide -\frac{10400}{179}, the coefficient of the x term, by 2 to get -\frac{5200}{179}. Then add the square of -\frac{5200}{179} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10400}{179}x+\left(-\frac{5200}{179}\right)^{2}=-\frac{80400}{179}+\left(-\frac{5200}{179}\right)^{2}$$

Square -\frac{5200}{179} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10400}{179}x+\frac{27040000}{32041}=-\frac{80400}{179}+\frac{27040000}{32041}$$

Add -\frac{80400}{179} to \frac{27040000}{32041} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{10400}{179}x+\frac{27040000}{32041}=\frac{12648400}{32041}$$

Factor x^{2}-\frac{10400}{179}x+\frac{27040000}{32041}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5200}{179}\right)^{2}=\frac{12648400}{32041}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5200}{179}\right)^{2}}=\sqrt{\frac{12648400}{32041}}$$

Simplify.

$$x-\frac{5200}{179}=\frac{20\sqrt{31621}}{179} x-\frac{5200}{179}=-\frac{20\sqrt{31621}}{179}$$

Add \frac{5200}{179} to both sides of the equation.

$$x=\frac{20\sqrt{31621}+5200}{179} x=\frac{5200-20\sqrt{31621}}{179}$$