Solve 100x^2+8x+6*3^2=5833

$$100 x ^ { 2 } + 8 x + 6 \cdot 3 ^ { 2 } = 5833$$

Solve for x

$$x=\frac{\sqrt{144479}}{50}-\frac{1}{25}\approx 7.562078663$$

$$x=-\frac{\sqrt{144479}}{50}-\frac{1}{25}\approx -7.642078663$$

Steps Using the Quadratic Formula

Calculate 3 to the power of 2 and get 9.

$$100x^{2}+8x+6\times 9=5833$$

Multiply 6 and 9 to get 54.

$$100x^{2}+8x+54=5833$$

Subtract 5833 from both sides.

$$100x^{2}+8x+54-5833=0$$

Subtract 5833 from 54 to get -5779.

$$100x^{2}+8x-5779=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 8 for b, and -5779 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 100\left(-5779\right)}}{2\times 100}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 100\left(-5779\right)}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-8±\sqrt{64-400\left(-5779\right)}}{2\times 100}$$

Multiply -400 times -5779.

$$x=\frac{-8±\sqrt{64+2311600}}{2\times 100}$$

Add 64 to 2311600.

$$x=\frac{-8±\sqrt{2311664}}{2\times 100}$$

Take the square root of 2311664.

$$x=\frac{-8±4\sqrt{144479}}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{-8±4\sqrt{144479}}{200}$$

Now solve the equation x=\frac{-8±4\sqrt{144479}}{200} when ± is plus. Add -8 to 4\sqrt{144479}.

$$x=\frac{4\sqrt{144479}-8}{200}$$

Divide -8+4\sqrt{144479} by 200.

$$x=\frac{\sqrt{144479}}{50}-\frac{1}{25}$$

Now solve the equation x=\frac{-8±4\sqrt{144479}}{200} when ± is minus. Subtract 4\sqrt{144479} from -8.

$$x=\frac{-4\sqrt{144479}-8}{200}$$

Divide -8-4\sqrt{144479} by 200.

$$x=-\frac{\sqrt{144479}}{50}-\frac{1}{25}$$

The equation is now solved.

$$x=\frac{\sqrt{144479}}{50}-\frac{1}{25} x=-\frac{\sqrt{144479}}{50}-\frac{1}{25}$$

Steps for Completing the Square

Calculate 3 to the power of 2 and get 9.

$$100x^{2}+8x+6\times 9=5833$$

Multiply 6 and 9 to get 54.

$$100x^{2}+8x+54=5833$$

Subtract 54 from both sides.

$$100x^{2}+8x=5833-54$$

Subtract 54 from 5833 to get 5779.

$$100x^{2}+8x=5779$$

Divide both sides by 100.

$$\frac{100x^{2}+8x}{100}=\frac{5779}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\frac{8}{100}x=\frac{5779}{100}$$

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

$$x^{2}+\frac{2}{25}x=\frac{5779}{100}$$

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.

$$x^{2}+\frac{2}{25}x+\left(\frac{1}{25}\right)^{2}=\frac{5779}{100}+\left(\frac{1}{25}\right)^{2}$$

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

$$x^{2}+\frac{2}{25}x+\frac{1}{625}=\frac{5779}{100}+\frac{1}{625}$$

Add \frac{5779}{100} to \frac{1}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{25}x+\frac{1}{625}=\frac{144479}{2500}$$

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}.

$$\left(x+\frac{1}{25}\right)^{2}=\frac{144479}{2500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{25}\right)^{2}}=\sqrt{\frac{144479}{2500}}$$

Simplify.

$$x+\frac{1}{25}=\frac{\sqrt{144479}}{50} x+\frac{1}{25}=-\frac{\sqrt{144479}}{50}$$

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

$$x=\frac{\sqrt{144479}}{50}-\frac{1}{25} x=-\frac{\sqrt{144479}}{50}-\frac{1}{25}$$