Solve y^2+6y-173=0

$${ y }^{ 2 } +6y-173 = 0$$

Solve for y (complex solution)

$$y=\sqrt{182}-3\approx 10.490737563$$

$$y=-\left(\sqrt{182}+3\right)\approx -16.490737563$$

Steps Using the Quadratic Formula

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.

$$y^{2}+6y-173=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -173 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-173\right)}}{2}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\left(-173\right)}}{2}$$

Multiply -4 times -173.

$$y=\frac{-6±\sqrt{36+692}}{2}$$

Add 36 to 692.

$$y=\frac{-6±\sqrt{728}}{2}$$

Take the square root of 728.

$$y=\frac{-6±2\sqrt{182}}{2}$$

Now solve the equation y=\frac{-6±2\sqrt{182}}{2} when ± is plus. Add -6 to 2\sqrt{182}.

$$y=\frac{2\sqrt{182}-6}{2}$$

Divide -6+2\sqrt{182} by 2.

$$y=\sqrt{182}-3$$

Now solve the equation y=\frac{-6±2\sqrt{182}}{2} when ± is minus. Subtract 2\sqrt{182} from -6.

$$y=\frac{-2\sqrt{182}-6}{2}$$

Divide -6-2\sqrt{182} by 2.

$$y=-\sqrt{182}-3$$

The equation is now solved.

$$y=\sqrt{182}-3 y=-\sqrt{182}-3$$

Steps for Completing the Square

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.

$$y^{2}+6y-173=0$$

Add 173 to both sides of the equation.

$$y^{2}+6y-173-\left(-173\right)=-\left(-173\right)$$

Subtracting -173 from itself leaves 0.

$$y^{2}+6y=-\left(-173\right)$$

Subtract -173 from 0.

$$y^{2}+6y=173$$

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.

$$y^{2}+6y+3^{2}=173+3^{2}$$

Square 3.

$$y^{2}+6y+9=173+9$$

Add 173 to 9.

$$y^{2}+6y+9=182$$

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}.

$$\left(y+3\right)^{2}=182$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{182}$$

Simplify.

$$y+3=\sqrt{182} y+3=-\sqrt{182}$$

Subtract 3 from both sides of the equation.

$$y=\sqrt{182}-3 y=-\sqrt{182}-3$$

Solve for y

$$y=\sqrt{182}-3\approx 10.490737563$$

$$y=-\sqrt{182}-3\approx -16.490737563$$

Steps Using the Quadratic Formula

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.

$$y^{2}+6y-173=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -173 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-173\right)}}{2}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\left(-173\right)}}{2}$$

Multiply -4 times -173.

$$y=\frac{-6±\sqrt{36+692}}{2}$$

Add 36 to 692.

$$y=\frac{-6±\sqrt{728}}{2}$$

Take the square root of 728.

$$y=\frac{-6±2\sqrt{182}}{2}$$

Now solve the equation y=\frac{-6±2\sqrt{182}}{2} when ± is plus. Add -6 to 2\sqrt{182}.

$$y=\frac{2\sqrt{182}-6}{2}$$

Divide -6+2\sqrt{182} by 2.

$$y=\sqrt{182}-3$$

Now solve the equation y=\frac{-6±2\sqrt{182}}{2} when ± is minus. Subtract 2\sqrt{182} from -6.

$$y=\frac{-2\sqrt{182}-6}{2}$$

Divide -6-2\sqrt{182} by 2.

$$y=-\sqrt{182}-3$$

The equation is now solved.

$$y=\sqrt{182}-3 y=-\sqrt{182}-3$$

Steps for Completing the Square

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.

$$y^{2}+6y-173=0$$

Add 173 to both sides of the equation.

$$y^{2}+6y-173-\left(-173\right)=-\left(-173\right)$$

Subtracting -173 from itself leaves 0.

$$y^{2}+6y=-\left(-173\right)$$

Subtract -173 from 0.

$$y^{2}+6y=173$$

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.

$$y^{2}+6y+3^{2}=173+3^{2}$$

Square 3.

$$y^{2}+6y+9=173+9$$

Add 173 to 9.

$$y^{2}+6y+9=182$$

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}.

$$\left(y+3\right)^{2}=182$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{182}$$

Simplify.

$$y+3=\sqrt{182} y+3=-\sqrt{182}$$

Subtract 3 from both sides of the equation.

$$y=\sqrt{182}-3 y=-\sqrt{182}-3$$