Solve 0.351=2x^2+.513x

$$0.351=2 { x }^{ 2 } +.513x$$

Solve for x

$$x=\frac{3\sqrt{341241}-513}{4000}\approx 0.309868777$$

$$x=\frac{-3\sqrt{341241}-513}{4000}\approx -0.566368777$$

Steps Using the Quadratic Formula

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

$$2x^{2}+0.513x=0.351$$

Subtract 0.351 from both sides.

$$2x^{2}+0.513x-0.351=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0.513 for b, and -0.351 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.513±\sqrt{0.513^{2}-4\times 2\left(-0.351\right)}}{2\times 2}$$

Square 0.513 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.513±\sqrt{0.263169-4\times 2\left(-0.351\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-0.513±\sqrt{0.263169-8\left(-0.351\right)}}{2\times 2}$$

Multiply -8 times -0.351.

$$x=\frac{-0.513±\sqrt{0.263169+2.808}}{2\times 2}$$

Add 0.263169 to 2.808 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.513±\sqrt{3.071169}}{2\times 2}$$

Take the square root of 3.071169.

$$x=\frac{-0.513±\frac{3\sqrt{341241}}{1000}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-0.513±\frac{3\sqrt{341241}}{1000}}{4}$$

Now solve the equation x=\frac{-0.513±\frac{3\sqrt{341241}}{1000}}{4} when ± is plus. Add -0.513 to \frac{3\sqrt{341241}}{1000}.

$$x=\frac{3\sqrt{341241}-513}{4\times 1000}$$

Divide \frac{-513+3\sqrt{341241}}{1000} by 4.

$$x=\frac{3\sqrt{341241}-513}{4000}$$

Now solve the equation x=\frac{-0.513±\frac{3\sqrt{341241}}{1000}}{4} when ± is minus. Subtract \frac{3\sqrt{341241}}{1000} from -0.513.

$$x=\frac{-3\sqrt{341241}-513}{4\times 1000}$$

Divide \frac{-513-3\sqrt{341241}}{1000} by 4.

$$x=\frac{-3\sqrt{341241}-513}{4000}$$

The equation is now solved.

$$x=\frac{3\sqrt{341241}-513}{4000} x=\frac{-3\sqrt{341241}-513}{4000}$$

Steps for Completing the Square

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

$$2x^{2}+0.513x=0.351$$

Divide both sides by 2.

$$\frac{2x^{2}+0.513x}{2}=\frac{0.351}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{0.513}{2}x=\frac{0.351}{2}$$

Divide 0.513 by 2.

$$x^{2}+0.2565x=\frac{0.351}{2}$$

Divide 0.351 by 2.

$$x^{2}+0.2565x=0.1755$$

Divide 0.2565, the coefficient of the x term, by 2 to get 0.12825. Then add the square of 0.12825 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.2565x+0.12825^{2}=0.1755+0.12825^{2}$$

Square 0.12825 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.2565x+0.0164480625=0.1755+0.0164480625$$

Add 0.1755 to 0.0164480625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.2565x+0.0164480625=0.1919480625$$

Factor x^{2}+0.2565x+0.0164480625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.12825\right)^{2}=0.1919480625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.12825\right)^{2}}=\sqrt{0.1919480625}$$

Simplify.

$$x+0.12825=\frac{3\sqrt{341241}}{4000} x+0.12825=-\frac{3\sqrt{341241}}{4000}$$

Subtract 0.12825 from both sides of the equation.

$$x=\frac{3\sqrt{341241}-513}{4000} x=\frac{-3\sqrt{341241}-513}{4000}$$