Solve (60x^2+32x+180)*0.4=103.2

$$(60 { x }^{ 2 } +32x+180) \times 0.4=103.2$$

Solve for x

$$x=\frac{\sqrt{1234}}{30}-\frac{4}{15}\approx 0.904277871$$

$$x=-\frac{\sqrt{1234}}{30}-\frac{4}{15}\approx -1.437611205$$

Steps Using the Quadratic Formula

Divide both sides by 0.4.

$$60x^{2}+32x+180=\frac{103.2}{0.4}$$

Expand \frac{103.2}{0.4} by multiplying both numerator and the denominator by 10.

$$60x^{2}+32x+180=\frac{1032}{4}$$

Divide 1032 by 4 to get 258.

$$60x^{2}+32x+180=258$$

Subtract 258 from both sides.

$$60x^{2}+32x+180-258=0$$

Subtract 258 from 180 to get -78.

$$60x^{2}+32x-78=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 60 for a, 32 for b, and -78 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\times 60\left(-78\right)}}{2\times 60}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\times 60\left(-78\right)}}{2\times 60}$$

Multiply -4 times 60.

$$x=\frac{-32±\sqrt{1024-240\left(-78\right)}}{2\times 60}$$

Multiply -240 times -78.

$$x=\frac{-32±\sqrt{1024+18720}}{2\times 60}$$

Add 1024 to 18720.

$$x=\frac{-32±\sqrt{19744}}{2\times 60}$$

Take the square root of 19744.

$$x=\frac{-32±4\sqrt{1234}}{2\times 60}$$

Multiply 2 times 60.

$$x=\frac{-32±4\sqrt{1234}}{120}$$

Now solve the equation x=\frac{-32±4\sqrt{1234}}{120} when ± is plus. Add -32 to 4\sqrt{1234}.

$$x=\frac{4\sqrt{1234}-32}{120}$$

Divide -32+4\sqrt{1234} by 120.

$$x=\frac{\sqrt{1234}}{30}-\frac{4}{15}$$

Now solve the equation x=\frac{-32±4\sqrt{1234}}{120} when ± is minus. Subtract 4\sqrt{1234} from -32.

$$x=\frac{-4\sqrt{1234}-32}{120}$$

Divide -32-4\sqrt{1234} by 120.

$$x=-\frac{\sqrt{1234}}{30}-\frac{4}{15}$$

The equation is now solved.

$$x=\frac{\sqrt{1234}}{30}-\frac{4}{15} x=-\frac{\sqrt{1234}}{30}-\frac{4}{15}$$

Steps for Completing the Square

Divide both sides by 0.4.

$$60x^{2}+32x+180=\frac{103.2}{0.4}$$

Expand \frac{103.2}{0.4} by multiplying both numerator and the denominator by 10.

$$60x^{2}+32x+180=\frac{1032}{4}$$

Divide 1032 by 4 to get 258.

$$60x^{2}+32x+180=258$$

Subtract 180 from both sides.

$$60x^{2}+32x=258-180$$

Subtract 180 from 258 to get 78.

$$60x^{2}+32x=78$$

Divide both sides by 60.

$$\frac{60x^{2}+32x}{60}=\frac{78}{60}$$

Dividing by 60 undoes the multiplication by 60.

$$x^{2}+\frac{32}{60}x=\frac{78}{60}$$

Reduce the fraction \frac{32}{60} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{8}{15}x=\frac{78}{60}$$

Reduce the fraction \frac{78}{60} to lowest terms by extracting and canceling out 6.

$$x^{2}+\frac{8}{15}x=\frac{13}{10}$$

Divide \frac{8}{15}, the coefficient of the x term, by 2 to get \frac{4}{15}. Then add the square of \frac{4}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{15}x+\left(\frac{4}{15}\right)^{2}=\frac{13}{10}+\left(\frac{4}{15}\right)^{2}$$

Square \frac{4}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{15}x+\frac{16}{225}=\frac{13}{10}+\frac{16}{225}$$

Add \frac{13}{10} to \frac{16}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{15}x+\frac{16}{225}=\frac{617}{450}$$

Factor x^{2}+\frac{8}{15}x+\frac{16}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4}{15}\right)^{2}=\frac{617}{450}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{15}\right)^{2}}=\sqrt{\frac{617}{450}}$$

Simplify.

$$x+\frac{4}{15}=\frac{\sqrt{1234}}{30} x+\frac{4}{15}=-\frac{\sqrt{1234}}{30}$$

Subtract \frac{4}{15} from both sides of the equation.

$$x=\frac{\sqrt{1234}}{30}-\frac{4}{15} x=-\frac{\sqrt{1234}}{30}-\frac{4}{15}$$