Solve 1.23=-0.0001x^2+0.0256x+0.5543

$$1.23 = -0.0001 { x }^{ 2 } +0.0256x+0.5543$$

Solve for x

$$x=\sqrt{9627}+128\approx 226.117276766$$

$$x=128-\sqrt{9627}\approx 29.882723234$$

Steps Using the Quadratic Formula

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

$$-0.0001x^{2}+0.0256x+0.5543=1.23$$

Subtract 1.23 from both sides.

$$-0.0001x^{2}+0.0256x+0.5543-1.23=0$$

Subtract 1.23 from 0.5543 to get -0.6757.

$$-0.0001x^{2}+0.0256x-0.6757=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.0001 for a, 0.0256 for b, and -0.6757 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.0256±\sqrt{0.0256^{2}-4\left(-0.0001\right)\left(-0.6757\right)}}{2\left(-0.0001\right)}$$

Square 0.0256 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.0256±\sqrt{0.00065536-4\left(-0.0001\right)\left(-0.6757\right)}}{2\left(-0.0001\right)}$$

Multiply -4 times -0.0001.

$$x=\frac{-0.0256±\sqrt{0.00065536+0.0004\left(-0.6757\right)}}{2\left(-0.0001\right)}$$

Multiply 0.0004 times -0.6757 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0256±\sqrt{0.00065536-0.00027028}}{2\left(-0.0001\right)}$$

Add 0.00065536 to -0.00027028 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.0256±\sqrt{0.00038508}}{2\left(-0.0001\right)}$$

Take the square root of 0.00038508.

$$x=\frac{-0.0256±\frac{\sqrt{9627}}{5000}}{2\left(-0.0001\right)}$$

Multiply 2 times -0.0001.

$$x=\frac{-0.0256±\frac{\sqrt{9627}}{5000}}{-0.0002}$$

Now solve the equation x=\frac{-0.0256±\frac{\sqrt{9627}}{5000}}{-0.0002} when ± is plus. Add -0.0256 to \frac{\sqrt{9627}}{5000}.

$$x=\frac{\frac{\sqrt{9627}}{5000}-\frac{16}{625}}{-0.0002}$$

Divide -\frac{16}{625}+\frac{\sqrt{9627}}{5000} by -0.0002 by multiplying -\frac{16}{625}+\frac{\sqrt{9627}}{5000} by the reciprocal of -0.0002.

$$x=128-\sqrt{9627}$$

Now solve the equation x=\frac{-0.0256±\frac{\sqrt{9627}}{5000}}{-0.0002} when ± is minus. Subtract \frac{\sqrt{9627}}{5000} from -0.0256.

$$x=\frac{-\frac{\sqrt{9627}}{5000}-\frac{16}{625}}{-0.0002}$$

Divide -\frac{16}{625}-\frac{\sqrt{9627}}{5000} by -0.0002 by multiplying -\frac{16}{625}-\frac{\sqrt{9627}}{5000} by the reciprocal of -0.0002.

$$x=\sqrt{9627}+128$$

The equation is now solved.

$$x=128-\sqrt{9627} x=\sqrt{9627}+128$$

Steps for Completing the Square

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

$$-0.0001x^{2}+0.0256x+0.5543=1.23$$

Subtract 0.5543 from both sides.

$$-0.0001x^{2}+0.0256x=1.23-0.5543$$

Subtract 0.5543 from 1.23 to get 0.6757.

$$-0.0001x^{2}+0.0256x=0.6757$$

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.

$$-0.0001x^{2}+0.0256x=\frac{6757}{10000}$$

Multiply both sides by -10000.

$$\frac{-0.0001x^{2}+0.0256x}{-0.0001}=\frac{\frac{6757}{10000}}{-0.0001}$$

Dividing by -0.0001 undoes the multiplication by -0.0001.

$$x^{2}+\frac{0.0256}{-0.0001}x=\frac{\frac{6757}{10000}}{-0.0001}$$

Divide 0.0256 by -0.0001 by multiplying 0.0256 by the reciprocal of -0.0001.

$$x^{2}-256x=\frac{\frac{6757}{10000}}{-0.0001}$$

Divide \frac{6757}{10000} by -0.0001 by multiplying \frac{6757}{10000} by the reciprocal of -0.0001.

$$x^{2}-256x=-6757$$

Divide -256, the coefficient of the x term, by 2 to get -128. Then add the square of -128 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-256x+\left(-128\right)^{2}=-6757+\left(-128\right)^{2}$$

Square -128.

$$x^{2}-256x+16384=-6757+16384$$

Add -6757 to 16384.

$$x^{2}-256x+16384=9627$$

Factor x^{2}-256x+16384. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-128\right)^{2}=9627$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-128\right)^{2}}=\sqrt{9627}$$

Simplify.

$$x-128=\sqrt{9627} x-128=-\sqrt{9627}$$

Add 128 to both sides of the equation.

$$x=\sqrt{9627}+128 x=128-\sqrt{9627}$$