Solve x^2-16.5x-19.6+0.1=0

$${ x }^{ 2 } -16.5x-19.6+0.1=0$$

Solve for x

$$x = \frac{\sqrt{1401} + 33}{4} \approx 17.607483636$$

$$x=\frac{33-\sqrt{1401}}{4}\approx -1.107483636$$

Steps Using the Quadratic Formula

Add -19.6 and 0.1 to get -19.5.

$$x^{2}-16.5x-19.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -16.5 for b, and -19.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16.5\right)±\sqrt{\left(-16.5\right)^{2}-4\left(-19.5\right)}}{2}$$

Square -16.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-16.5\right)±\sqrt{272.25-4\left(-19.5\right)}}{2}$$

Multiply -4 times -19.5.

$$x=\frac{-\left(-16.5\right)±\sqrt{272.25+78}}{2}$$

Add 272.25 to 78.

$$x=\frac{-\left(-16.5\right)±\sqrt{350.25}}{2}$$

Take the square root of 350.25.

$$x=\frac{-\left(-16.5\right)±\frac{\sqrt{1401}}{2}}{2}$$

The opposite of -16.5 is 16.5.

$$x=\frac{16.5±\frac{\sqrt{1401}}{2}}{2}$$

Now solve the equation x=\frac{16.5±\frac{\sqrt{1401}}{2}}{2} when ± is plus. Add 16.5 to \frac{\sqrt{1401}}{2}.

$$x=\frac{\sqrt{1401}+33}{2\times 2}$$

Divide \frac{33+\sqrt{1401}}{2} by 2.

$$x=\frac{\sqrt{1401}+33}{4}$$

Now solve the equation x=\frac{16.5±\frac{\sqrt{1401}}{2}}{2} when ± is minus. Subtract \frac{\sqrt{1401}}{2} from 16.5.

$$x=\frac{33-\sqrt{1401}}{2\times 2}$$

Divide \frac{33-\sqrt{1401}}{2} by 2.

$$x=\frac{33-\sqrt{1401}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{1401}+33}{4} x=\frac{33-\sqrt{1401}}{4}$$

Steps for Completing the Square

Add -19.6 and 0.1 to get -19.5.

$$x^{2}-16.5x-19.5=0$$

Add 19.5 to both sides. Anything plus zero gives itself.

$$x^{2}-16.5x=19.5$$

Divide -16.5, the coefficient of the x term, by 2 to get -8.25. Then add the square of -8.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16.5x+\left(-8.25\right)^{2}=19.5+\left(-8.25\right)^{2}$$

Square -8.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-16.5x+68.0625=19.5+68.0625$$

Add 19.5 to 68.0625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-16.5x+68.0625=87.5625$$

Factor x^{2}-16.5x+68.0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8.25\right)^{2}=87.5625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8.25\right)^{2}}=\sqrt{87.5625}$$

Simplify.

$$x-8.25=\frac{\sqrt{1401}}{4} x-8.25=-\frac{\sqrt{1401}}{4}$$

Add 8.25 to both sides of the equation.

$$x=\frac{\sqrt{1401}+33}{4} x=\frac{33-\sqrt{1401}}{4}$$