Solve 2x^2=-3.5x-1

$$2 x ^ { 2 } = - 3.5 x - 1$$

Solve for x

$$x=\frac{\sqrt{17}-7}{8}\approx -0.359611797$$

$$x=\frac{-\sqrt{17}-7}{8}\approx -1.390388203$$

Steps Using the Quadratic Formula

Add 3.5x to both sides.

$$2x^{2}+3.5x=-1$$

Add 1 to both sides.

$$2x^{2}+3.5x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 3.5 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3.5±\sqrt{3.5^{2}-4\times 2}}{2\times 2}$$

Square 3.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-3.5±\sqrt{12.25-4\times 2}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-3.5±\sqrt{12.25-8}}{2\times 2}$$

Add 12.25 to -8.

$$x=\frac{-3.5±\sqrt{4.25}}{2\times 2}$$

Take the square root of 4.25.

$$x=\frac{-3.5±\frac{\sqrt{17}}{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-3.5±\frac{\sqrt{17}}{2}}{4}$$

Now solve the equation x=\frac{-3.5±\frac{\sqrt{17}}{2}}{4} when ± is plus. Add -3.5 to \frac{\sqrt{17}}{2}.

$$x=\frac{\sqrt{17}-7}{2\times 4}$$

Divide \frac{-7+\sqrt{17}}{2} by 4.

$$x=\frac{\sqrt{17}-7}{8}$$

Now solve the equation x=\frac{-3.5±\frac{\sqrt{17}}{2}}{4} when ± is minus. Subtract \frac{\sqrt{17}}{2} from -3.5.

$$x=\frac{-\sqrt{17}-7}{2\times 4}$$

Divide \frac{-7-\sqrt{17}}{2} by 4.

$$x=\frac{-\sqrt{17}-7}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{17}-7}{8} x=\frac{-\sqrt{17}-7}{8}$$

Steps for Completing the Square

Add 3.5x to both sides.

$$2x^{2}+3.5x=-1$$

Divide both sides by 2.

$$\frac{2x^{2}+3.5x}{2}=-\frac{1}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{3.5}{2}x=-\frac{1}{2}$$

Divide 3.5 by 2.

$$x^{2}+1.75x=-\frac{1}{2}$$

Divide 1.75, the coefficient of the x term, by 2 to get 0.875. Then add the square of 0.875 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+1.75x+0.875^{2}=-\frac{1}{2}+0.875^{2}$$

Square 0.875 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+1.75x+0.765625=-\frac{1}{2}+0.765625$$

Add -\frac{1}{2} to 0.765625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+1.75x+0.765625=\frac{17}{64}$$

Factor x^{2}+1.75x+0.765625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.875\right)^{2}=\frac{17}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.875\right)^{2}}=\sqrt{\frac{17}{64}}$$

Simplify.

$$x+0.875=\frac{\sqrt{17}}{8} x+0.875=-\frac{\sqrt{17}}{8}$$

Subtract 0.875 from both sides of the equation.

$$x=\frac{\sqrt{17}-7}{8} x=\frac{-\sqrt{17}-7}{8}$$