Solve yy-3-+y=22

$$y y - 3 - + y = 22$$

Solve for y

$$y = \frac{\sqrt{101} + 1}{2} \approx 5.524937811$$

$$y=\frac{1-\sqrt{101}}{2}\approx -4.524937811$$

Steps Using the Quadratic Formula

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

$$y^{2}-3-y=22$$

Subtract 22 from both sides.

$$y^{2}-3-y-22=0$$

Subtract 22 from -3 to get -25.

$$y^{2}-25-y=0$$

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.

$$y^{2}-y-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1-4\left(-25\right)}}{2}$$

Multiply -4 times -25.

$$y=\frac{-\left(-1\right)±\sqrt{1+100}}{2}$$

Add 1 to 100.

$$y=\frac{-\left(-1\right)±\sqrt{101}}{2}$$

The opposite of -1 is 1.

$$y=\frac{1±\sqrt{101}}{2}$$

Now solve the equation y=\frac{1±\sqrt{101}}{2} when ± is plus. Add 1 to \sqrt{101}.

$$y=\frac{\sqrt{101}+1}{2}$$

Now solve the equation y=\frac{1±\sqrt{101}}{2} when ± is minus. Subtract \sqrt{101} from 1.

$$y=\frac{1-\sqrt{101}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{101}+1}{2} y=\frac{1-\sqrt{101}}{2}$$

Steps for Completing the Square

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

$$y^{2}-3-y=22$$

Add 3 to both sides.

$$y^{2}-y=22+3$$

Add 22 and 3 to get 25.

$$y^{2}-y=25$$

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.

$$y^{2}-y+\left(-\frac{1}{2}\right)^{2}=25+\left(-\frac{1}{2}\right)^{2}$$

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

$$y^{2}-y+\frac{1}{4}=25+\frac{1}{4}$$

Add 25 to \frac{1}{4}.

$$y^{2}-y+\frac{1}{4}=\frac{101}{4}$$

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}.

$$\left(y-\frac{1}{2}\right)^{2}=\frac{101}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{2}\right)^{2}}=\sqrt{\frac{101}{4}}$$

Simplify.

$$y-\frac{1}{2}=\frac{\sqrt{101}}{2} y-\frac{1}{2}=-\frac{\sqrt{101}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$y=\frac{\sqrt{101}+1}{2} y=\frac{1-\sqrt{101}}{2}$$