Solve y+y*y=9

$$y + y \cdot y = 9$$

Solve for y

$$y = \frac{\sqrt{37} - 1}{2} \approx 2.541381265$$

$$y=\frac{-\sqrt{37}-1}{2}\approx -3.541381265$$

Steps Using the Quadratic Formula

Multiply y and y to get y^{2}.

$$y+y^{2}=9$$

Subtract 9 from both sides.

$$y+y^{2}-9=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$y^{2}+y-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-1±\sqrt{1^{2}-4\left(-9\right)}}{2}$$

Square 1.

$$y=\frac{-1±\sqrt{1-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$y=\frac{-1±\sqrt{1+36}}{2}$$

Add 1 to 36.

$$y=\frac{-1±\sqrt{37}}{2}$$

Now solve the equation y=\frac{-1±\sqrt{37}}{2} when ± is plus. Add -1 to \sqrt{37}.

$$y=\frac{\sqrt{37}-1}{2}$$

Now solve the equation y=\frac{-1±\sqrt{37}}{2} when ± is minus. Subtract \sqrt{37} from -1.

$$y=\frac{-\sqrt{37}-1}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{37}-1}{2} y=\frac{-\sqrt{37}-1}{2}$$

Steps for Completing the Square

Multiply y and y to get y^{2}.

$$y+y^{2}=9$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$y^{2}+y=9$$

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

$$y^{2}+y+\left(\frac{1}{2}\right)^{2}=9+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+y+\frac{1}{4}=9+\frac{1}{4}$$

Add 9 to \frac{1}{4}.

$$y^{2}+y+\frac{1}{4}=\frac{37}{4}$$

Factor y^{2}+y+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{1}{2}\right)^{2}=\frac{37}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{1}{2}\right)^{2}}=\sqrt{\frac{37}{4}}$$

Simplify.

$$y+\frac{1}{2}=\frac{\sqrt{37}}{2} y+\frac{1}{2}=-\frac{\sqrt{37}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$y=\frac{\sqrt{37}-1}{2} y=\frac{-\sqrt{37}-1}{2}$$