Solve -2y-4+(y-2)^2=20

$$- 2 y - 4 + ( y - 2 ) ^ { 2 } = 20$$

Solve for y

$$y=\sqrt{29}+3\approx 8.385164807$$

$$y=3-\sqrt{29}\approx -2.385164807$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-2\right)^{2}.

$$-2y-4+y^{2}-4y+4=20$$

Combine -2y and -4y to get -6y.

$$-6y-4+y^{2}+4=20$$

Add -4 and 4 to get 0.

$$-6y+y^{2}=20$$

Subtract 20 from both sides.

$$-6y+y^{2}-20=0$$

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.

$$y^{2}-6y-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-20\right)}}{2}$$

Square -6.

$$y=\frac{-\left(-6\right)±\sqrt{36-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$y=\frac{-\left(-6\right)±\sqrt{36+80}}{2}$$

Add 36 to 80.

$$y=\frac{-\left(-6\right)±\sqrt{116}}{2}$$

Take the square root of 116.

$$y=\frac{-\left(-6\right)±2\sqrt{29}}{2}$$

The opposite of -6 is 6.

$$y=\frac{6±2\sqrt{29}}{2}$$

Now solve the equation y=\frac{6±2\sqrt{29}}{2} when ± is plus. Add 6 to 2\sqrt{29}.

$$y=\frac{2\sqrt{29}+6}{2}$$

Divide 6+2\sqrt{29} by 2.

$$y=\sqrt{29}+3$$

Now solve the equation y=\frac{6±2\sqrt{29}}{2} when ± is minus. Subtract 2\sqrt{29} from 6.

$$y=\frac{6-2\sqrt{29}}{2}$$

Divide 6-2\sqrt{29} by 2.

$$y=3-\sqrt{29}$$

The equation is now solved.

$$y=\sqrt{29}+3 y=3-\sqrt{29}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-2\right)^{2}.

$$-2y-4+y^{2}-4y+4=20$$

Combine -2y and -4y to get -6y.

$$-6y-4+y^{2}+4=20$$

Add -4 and 4 to get 0.

$$-6y+y^{2}=20$$

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.

$$y^{2}-6y=20$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-6y+\left(-3\right)^{2}=20+\left(-3\right)^{2}$$

Square -3.

$$y^{2}-6y+9=20+9$$

Add 20 to 9.

$$y^{2}-6y+9=29$$

Factor y^{2}-6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-3\right)^{2}=29$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{29}$$

Simplify.

$$y-3=\sqrt{29} y-3=-\sqrt{29}$$

Add 3 to both sides of the equation.

$$y=\sqrt{29}+3 y=3-\sqrt{29}$$