Solve 0=(-0.000234)left(x-80right)^2+1.5

$$0=(-0.000234) { \left(x-80 \right) }^{ 2 } +1.5$$

Solve for x

$$x=\frac{500\sqrt{39}}{39}+80\approx 160.064076903$$

$$x=-\frac{500\sqrt{39}}{39}+80\approx -0.064076903$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-80\right)^{2}.

$$0=-0.000234\left(x^{2}-160x+6400\right)+1.5$$

Use the distributive property to multiply -0.000234 by x^{2}-160x+6400.

$$0=-0.000234x^{2}+0.03744x-1.4976+1.5$$

Add -1.4976 and 1.5 to get 0.0024.

$$0=-0.000234x^{2}+0.03744x+0.0024$$

Swap sides so that all variable terms are on the left hand side.

$$-0.000234x^{2}+0.03744x+0.0024=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.000234 for a, 0.03744 for b, and 0.0024 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.03744±\sqrt{0.03744^{2}-4\left(-0.000234\right)\times 0.0024}}{2\left(-0.000234\right)}$$

Square 0.03744 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.03744±\sqrt{0.0014017536-4\left(-0.000234\right)\times 0.0024}}{2\left(-0.000234\right)}$$

Multiply -4 times -0.000234.

$$x=\frac{-0.03744±\sqrt{0.0014017536+0.000936\times 0.0024}}{2\left(-0.000234\right)}$$

Multiply 0.000936 times 0.0024 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.03744±\sqrt{0.0014017536+0.0000022464}}{2\left(-0.000234\right)}$$

Add 0.0014017536 to 0.0000022464 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.03744±\sqrt{0.001404}}{2\left(-0.000234\right)}$$

Take the square root of 0.001404.

$$x=\frac{-0.03744±\frac{3\sqrt{39}}{500}}{2\left(-0.000234\right)}$$

Multiply 2 times -0.000234.

$$x=\frac{-0.03744±\frac{3\sqrt{39}}{500}}{-0.000468}$$

Now solve the equation x=\frac{-0.03744±\frac{3\sqrt{39}}{500}}{-0.000468} when ± is plus. Add -0.03744 to \frac{3\sqrt{39}}{500}.

$$x=\frac{\frac{3\sqrt{39}}{500}-\frac{117}{3125}}{-0.000468}$$

Divide -\frac{117}{3125}+\frac{3\sqrt{39}}{500} by -0.000468 by multiplying -\frac{117}{3125}+\frac{3\sqrt{39}}{500} by the reciprocal of -0.000468.

$$x=-\frac{500\sqrt{39}}{39}+80$$

Now solve the equation x=\frac{-0.03744±\frac{3\sqrt{39}}{500}}{-0.000468} when ± is minus. Subtract \frac{3\sqrt{39}}{500} from -0.03744.

$$x=\frac{-\frac{3\sqrt{39}}{500}-\frac{117}{3125}}{-0.000468}$$

Divide -\frac{117}{3125}-\frac{3\sqrt{39}}{500} by -0.000468 by multiplying -\frac{117}{3125}-\frac{3\sqrt{39}}{500} by the reciprocal of -0.000468.

$$x=\frac{500\sqrt{39}}{39}+80$$

The equation is now solved.

$$x=-\frac{500\sqrt{39}}{39}+80 x=\frac{500\sqrt{39}}{39}+80$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-80\right)^{2}.

$$0=-0.000234\left(x^{2}-160x+6400\right)+1.5$$

Use the distributive property to multiply -0.000234 by x^{2}-160x+6400.

$$0=-0.000234x^{2}+0.03744x-1.4976+1.5$$

Add -1.4976 and 1.5 to get 0.0024.

$$0=-0.000234x^{2}+0.03744x+0.0024$$

Swap sides so that all variable terms are on the left hand side.

$$-0.000234x^{2}+0.03744x+0.0024=0$$

Subtract 0.0024 from both sides. Anything subtracted from zero gives its negation.

$$-0.000234x^{2}+0.03744x=-0.0024$$

Divide both sides of the equation by -0.000234, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.000234x^{2}+0.03744x}{-0.000234}=-\frac{0.0024}{-0.000234}$$

Dividing by -0.000234 undoes the multiplication by -0.000234.

$$x^{2}+\frac{0.03744}{-0.000234}x=-\frac{0.0024}{-0.000234}$$

Divide 0.03744 by -0.000234 by multiplying 0.03744 by the reciprocal of -0.000234.

$$x^{2}-160x=-\frac{0.0024}{-0.000234}$$

Divide -0.0024 by -0.000234 by multiplying -0.0024 by the reciprocal of -0.000234.

$$x^{2}-160x=\frac{400}{39}$$

Divide -160, the coefficient of the x term, by 2 to get -80. Then add the square of -80 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-160x+\left(-80\right)^{2}=\frac{400}{39}+\left(-80\right)^{2}$$

Square -80.

$$x^{2}-160x+6400=\frac{400}{39}+6400$$

Add \frac{400}{39} to 6400.

$$x^{2}-160x+6400=\frac{250000}{39}$$

Factor x^{2}-160x+6400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-80\right)^{2}=\frac{250000}{39}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-80\right)^{2}}=\sqrt{\frac{250000}{39}}$$

Simplify.

$$x-80=\frac{500\sqrt{39}}{39} x-80=-\frac{500\sqrt{39}}{39}$$

Add 80 to both sides of the equation.

$$x=\frac{500\sqrt{39}}{39}+80 x=-\frac{500\sqrt{39}}{39}+80$$