Solve gamma^2+86gamma-60=0

$$\gamma ^ { 2 } + 86 \gamma - 60 = 0$$

Solve for γ (complex solution)

$$\gamma =\sqrt{1909}-43\approx 0.69210455$$

$$\gamma =-\left(\sqrt{1909}+43\right)\approx -86.69210455$$

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.

$$\gamma ^{2}+86\gamma -60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 86 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\gamma =\frac{-86±\sqrt{86^{2}-4\left(-60\right)}}{2}$$

Square 86.

$$\gamma =\frac{-86±\sqrt{7396-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$\gamma =\frac{-86±\sqrt{7396+240}}{2}$$

Add 7396 to 240.

$$\gamma =\frac{-86±\sqrt{7636}}{2}$$

Take the square root of 7636.

$$\gamma =\frac{-86±2\sqrt{1909}}{2}$$

Now solve the equation \gamma =\frac{-86±2\sqrt{1909}}{2} when ± is plus. Add -86 to 2\sqrt{1909}.

$$\gamma =\frac{2\sqrt{1909}-86}{2}$$

Divide -86+2\sqrt{1909} by 2.

$$\gamma =\sqrt{1909}-43$$

Now solve the equation \gamma =\frac{-86±2\sqrt{1909}}{2} when ± is minus. Subtract 2\sqrt{1909} from -86.

$$\gamma =\frac{-2\sqrt{1909}-86}{2}$$

Divide -86-2\sqrt{1909} by 2.

$$\gamma =-\sqrt{1909}-43$$

The equation is now solved.

$$\gamma =\sqrt{1909}-43 \gamma =-\sqrt{1909}-43$$

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.

$$\gamma ^{2}+86\gamma -60=0$$

Add 60 to both sides of the equation.

$$\gamma ^{2}+86\gamma -60-\left(-60\right)=-\left(-60\right)$$

Subtracting -60 from itself leaves 0.

$$\gamma ^{2}+86\gamma =-\left(-60\right)$$

Subtract -60 from 0.

$$\gamma ^{2}+86\gamma =60$$

Divide 86, the coefficient of the x term, by 2 to get 43. Then add the square of 43 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\gamma ^{2}+86\gamma +43^{2}=60+43^{2}$$

Square 43.

$$\gamma ^{2}+86\gamma +1849=60+1849$$

Add 60 to 1849.

$$\gamma ^{2}+86\gamma +1849=1909$$

Factor \gamma ^{2}+86\gamma +1849. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\gamma +43\right)^{2}=1909$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\gamma +43\right)^{2}}=\sqrt{1909}$$

Simplify.

$$\gamma +43=\sqrt{1909} \gamma +43=-\sqrt{1909}$$

Subtract 43 from both sides of the equation.

$$\gamma =\sqrt{1909}-43 \gamma =-\sqrt{1909}-43$$

Solve for γ

$$\gamma =\sqrt{1909}-43\approx 0.69210455$$

$$\gamma =-\sqrt{1909}-43\approx -86.69210455$$

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.

$$\gamma ^{2}+86\gamma -60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 86 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\gamma =\frac{-86±\sqrt{86^{2}-4\left(-60\right)}}{2}$$

Square 86.

$$\gamma =\frac{-86±\sqrt{7396-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$\gamma =\frac{-86±\sqrt{7396+240}}{2}$$

Add 7396 to 240.

$$\gamma =\frac{-86±\sqrt{7636}}{2}$$

Take the square root of 7636.

$$\gamma =\frac{-86±2\sqrt{1909}}{2}$$

Now solve the equation \gamma =\frac{-86±2\sqrt{1909}}{2} when ± is plus. Add -86 to 2\sqrt{1909}.

$$\gamma =\frac{2\sqrt{1909}-86}{2}$$

Divide -86+2\sqrt{1909} by 2.

$$\gamma =\sqrt{1909}-43$$

Now solve the equation \gamma =\frac{-86±2\sqrt{1909}}{2} when ± is minus. Subtract 2\sqrt{1909} from -86.

$$\gamma =\frac{-2\sqrt{1909}-86}{2}$$

Divide -86-2\sqrt{1909} by 2.

$$\gamma =-\sqrt{1909}-43$$

The equation is now solved.

$$\gamma =\sqrt{1909}-43 \gamma =-\sqrt{1909}-43$$

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.

$$\gamma ^{2}+86\gamma -60=0$$

Add 60 to both sides of the equation.

$$\gamma ^{2}+86\gamma -60-\left(-60\right)=-\left(-60\right)$$

Subtracting -60 from itself leaves 0.

$$\gamma ^{2}+86\gamma =-\left(-60\right)$$

Subtract -60 from 0.

$$\gamma ^{2}+86\gamma =60$$

Divide 86, the coefficient of the x term, by 2 to get 43. Then add the square of 43 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\gamma ^{2}+86\gamma +43^{2}=60+43^{2}$$

Square 43.

$$\gamma ^{2}+86\gamma +1849=60+1849$$

Add 60 to 1849.

$$\gamma ^{2}+86\gamma +1849=1909$$

Factor \gamma ^{2}+86\gamma +1849. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\gamma +43\right)^{2}=1909$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\gamma +43\right)^{2}}=\sqrt{1909}$$

Simplify.

$$\gamma +43=\sqrt{1909} \gamma +43=-\sqrt{1909}$$

Subtract 43 from both sides of the equation.

$$\gamma =\sqrt{1909}-43 \gamma =-\sqrt{1909}-43$$