Solve 100x^2+8x+6*3^2=583.3

$$100 x ^ { 2 } + 8 x + 6 \cdot 3 ^ { 2 } = 583.3$$

Solve for x

$$x=\frac{\sqrt{52946}}{100}-0.04\approx 2.260999783$$

$$x=-\frac{\sqrt{52946}}{100}-0.04\approx -2.340999783$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 2 and get 9.

$$100x^{2}+8x+6\times 9=583.3$$

Multiply 6 and 9 to get 54.

$$100x^{2}+8x+54=583.3$$

Subtract 583.3 from both sides.

$$100x^{2}+8x+54-583.3=0$$

Subtract 583.3 from 54 to get -529.3.

$$100x^{2}+8x-529.3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 8 for b, and -529.3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 100\left(-529.3\right)}}{2\times 100}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 100\left(-529.3\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-8±\sqrt{64-400\left(-529.3\right)}}{2\times 100}$$

Multiply -400 times -529.3.

$$x=\frac{-8±\sqrt{64+211720}}{2\times 100}$$

Add 64 to 211720.

$$x=\frac{-8±\sqrt{211784}}{2\times 100}$$

Take the square root of 211784.

$$x=\frac{-8±2\sqrt{52946}}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{-8±2\sqrt{52946}}{200}$$

Now solve the equation x=\frac{-8±2\sqrt{52946}}{200} when ± is plus. Add -8 to 2\sqrt{52946}.

$$x=\frac{2\sqrt{52946}-8}{200}$$

Divide -8+2\sqrt{52946} by 200.

$$x=\frac{\sqrt{52946}}{100}-\frac{1}{25}$$

Now solve the equation x=\frac{-8±2\sqrt{52946}}{200} when ± is minus. Subtract 2\sqrt{52946} from -8.

$$x=\frac{-2\sqrt{52946}-8}{200}$$

Divide -8-2\sqrt{52946} by 200.

$$x=-\frac{\sqrt{52946}}{100}-\frac{1}{25}$$

The equation is now solved.

$$x=\frac{\sqrt{52946}}{100}-\frac{1}{25} x=-\frac{\sqrt{52946}}{100}-\frac{1}{25}$$

Steps for Completing the Square

Calculate 3 to the power of 2 and get 9.

$$100x^{2}+8x+6\times 9=583.3$$

Multiply 6 and 9 to get 54.

$$100x^{2}+8x+54=583.3$$

Subtract 54 from both sides.

$$100x^{2}+8x=583.3-54$$

Subtract 54 from 583.3 to get 529.3.

$$100x^{2}+8x=529.3$$

Divide both sides by 100.

$$\frac{100x^{2}+8x}{100}=\frac{529.3}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\frac{8}{100}x=\frac{529.3}{100}$$

Reduce the fraction \frac{8}{100} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{2}{25}x=\frac{529.3}{100}$$

Divide 529.3 by 100.

$$x^{2}+\frac{2}{25}x=5.293$$

Divide \frac{2}{25}, the coefficient of the x term, by 2 to get \frac{1}{25}. Then add the square of \frac{1}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{25}x+\left(\frac{1}{25}\right)^{2}=5.293+\left(\frac{1}{25}\right)^{2}$$

Square \frac{1}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{25}x+\frac{1}{625}=5.293+\frac{1}{625}$$

Add 5.293 to \frac{1}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{25}x+\frac{1}{625}=\frac{26473}{5000}$$

Factor x^{2}+\frac{2}{25}x+\frac{1}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{25}\right)^{2}=\frac{26473}{5000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{25}\right)^{2}}=\sqrt{\frac{26473}{5000}}$$

Simplify.

$$x+\frac{1}{25}=\frac{\sqrt{52946}}{100} x+\frac{1}{25}=-\frac{\sqrt{52946}}{100}$$

Subtract \frac{1}{25} from both sides of the equation.

$$x=\frac{\sqrt{52946}}{100}-\frac{1}{25} x=-\frac{\sqrt{52946}}{100}-\frac{1}{25}$$