Solve y(2y+1)+10=0

$$y ( 2 y + 1 ) + 10 = 0$$

Solve for y

$$y=\frac{-1+\sqrt{79}i}{4}\approx -0.25+2.222048604i$$

$$y=\frac{-\sqrt{79}i-1}{4}\approx -0.25-2.222048604i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply y by 2y+1.

$$2y^{2}+y+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 1 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-1±\sqrt{1^{2}-4\times 2\times 10}}{2\times 2}$$

Square 1.

$$y=\frac{-1±\sqrt{1-4\times 2\times 10}}{2\times 2}$$

Multiply -4 times 2.

$$y=\frac{-1±\sqrt{1-8\times 10}}{2\times 2}$$

Multiply -8 times 10.

$$y=\frac{-1±\sqrt{1-80}}{2\times 2}$$

Add 1 to -80.

$$y=\frac{-1±\sqrt{-79}}{2\times 2}$$

Take the square root of -79.

$$y=\frac{-1±\sqrt{79}i}{2\times 2}$$

Multiply 2 times 2.

$$y=\frac{-1±\sqrt{79}i}{4}$$

Now solve the equation y=\frac{-1±\sqrt{79}i}{4} when ± is plus. Add -1 to i\sqrt{79}.

$$y=\frac{-1+\sqrt{79}i}{4}$$

Now solve the equation y=\frac{-1±\sqrt{79}i}{4} when ± is minus. Subtract i\sqrt{79} from -1.

$$y=\frac{-\sqrt{79}i-1}{4}$$

The equation is now solved.

$$y=\frac{-1+\sqrt{79}i}{4} y=\frac{-\sqrt{79}i-1}{4}$$

Steps for Completing the Square

Use the distributive property to multiply y by 2y+1.

$$2y^{2}+y+10=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$2y^{2}+y=-10$$

Divide both sides by 2.

$$\frac{2y^{2}+y}{2}=-\frac{10}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$y^{2}+\frac{1}{2}y=-\frac{10}{2}$$

Divide -10 by 2.

$$y^{2}+\frac{1}{2}y=-5$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{1}{2}y+\left(\frac{1}{4}\right)^{2}=-5+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{1}{2}y+\frac{1}{16}=-5+\frac{1}{16}$$

Add -5 to \frac{1}{16}.

$$y^{2}+\frac{1}{2}y+\frac{1}{16}=-\frac{79}{16}$$

Factor y^{2}+\frac{1}{2}y+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{1}{4}\right)^{2}=-\frac{79}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{1}{4}\right)^{2}}=\sqrt{-\frac{79}{16}}$$

Simplify.

$$y+\frac{1}{4}=\frac{\sqrt{79}i}{4} y+\frac{1}{4}=-\frac{\sqrt{79}i}{4}$$

Subtract \frac{1}{4} from both sides of the equation.

$$y=\frac{-1+\sqrt{79}i}{4} y=\frac{-\sqrt{79}i-1}{4}$$