Solve x^2+74x-6977=0

$${ x }^{ 2 } +74x-6977=0$$

Solve for x (complex solution)

$$x=\sqrt{8346}-37\approx 54.356444764$$

$$x=-\left(\sqrt{8346}+37\right)\approx -128.356444764$$

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.

$$x^{2}+74x-6977=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 74 for b, and -6977 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\left(-6977\right)}}{2}$$

Square 74.

$$x=\frac{-74±\sqrt{5476-4\left(-6977\right)}}{2}$$

Multiply -4 times -6977.

$$x=\frac{-74±\sqrt{5476+27908}}{2}$$

Add 5476 to 27908.

$$x=\frac{-74±\sqrt{33384}}{2}$$

Take the square root of 33384.

$$x=\frac{-74±2\sqrt{8346}}{2}$$

Now solve the equation x=\frac{-74±2\sqrt{8346}}{2} when ± is plus. Add -74 to 2\sqrt{8346}.

$$x=\frac{2\sqrt{8346}-74}{2}$$

Divide -74+2\sqrt{8346} by 2.

$$x=\sqrt{8346}-37$$

Now solve the equation x=\frac{-74±2\sqrt{8346}}{2} when ± is minus. Subtract 2\sqrt{8346} from -74.

$$x=\frac{-2\sqrt{8346}-74}{2}$$

Divide -74-2\sqrt{8346} by 2.

$$x=-\sqrt{8346}-37$$

The equation is now solved.

$$x=\sqrt{8346}-37 x=-\sqrt{8346}-37$$

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.

$$x^{2}+74x-6977=0$$

Add 6977 to both sides of the equation.

$$x^{2}+74x-6977-\left(-6977\right)=-\left(-6977\right)$$

Subtracting -6977 from itself leaves 0.

$$x^{2}+74x=-\left(-6977\right)$$

Subtract -6977 from 0.

$$x^{2}+74x=6977$$

Divide 74, the coefficient of the x term, by 2 to get 37. Then add the square of 37 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+74x+37^{2}=6977+37^{2}$$

Square 37.

$$x^{2}+74x+1369=6977+1369$$

Add 6977 to 1369.

$$x^{2}+74x+1369=8346$$

Factor x^{2}+74x+1369. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+37\right)^{2}=8346$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+37\right)^{2}}=\sqrt{8346}$$

Simplify.

$$x+37=\sqrt{8346} x+37=-\sqrt{8346}$$

Subtract 37 from both sides of the equation.

$$x=\sqrt{8346}-37 x=-\sqrt{8346}-37$$

Solve for x

$$x=\sqrt{8346}-37\approx 54.356444764$$

$$x=-\sqrt{8346}-37\approx -128.356444764$$

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.

$$x^{2}+74x-6977=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 74 for b, and -6977 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\left(-6977\right)}}{2}$$

Square 74.

$$x=\frac{-74±\sqrt{5476-4\left(-6977\right)}}{2}$$

Multiply -4 times -6977.

$$x=\frac{-74±\sqrt{5476+27908}}{2}$$

Add 5476 to 27908.

$$x=\frac{-74±\sqrt{33384}}{2}$$

Take the square root of 33384.

$$x=\frac{-74±2\sqrt{8346}}{2}$$

Now solve the equation x=\frac{-74±2\sqrt{8346}}{2} when ± is plus. Add -74 to 2\sqrt{8346}.

$$x=\frac{2\sqrt{8346}-74}{2}$$

Divide -74+2\sqrt{8346} by 2.

$$x=\sqrt{8346}-37$$

Now solve the equation x=\frac{-74±2\sqrt{8346}}{2} when ± is minus. Subtract 2\sqrt{8346} from -74.

$$x=\frac{-2\sqrt{8346}-74}{2}$$

Divide -74-2\sqrt{8346} by 2.

$$x=-\sqrt{8346}-37$$

The equation is now solved.

$$x=\sqrt{8346}-37 x=-\sqrt{8346}-37$$

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.

$$x^{2}+74x-6977=0$$

Add 6977 to both sides of the equation.

$$x^{2}+74x-6977-\left(-6977\right)=-\left(-6977\right)$$

Subtracting -6977 from itself leaves 0.

$$x^{2}+74x=-\left(-6977\right)$$

Subtract -6977 from 0.

$$x^{2}+74x=6977$$

Divide 74, the coefficient of the x term, by 2 to get 37. Then add the square of 37 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+74x+37^{2}=6977+37^{2}$$

Square 37.

$$x^{2}+74x+1369=6977+1369$$

Add 6977 to 1369.

$$x^{2}+74x+1369=8346$$

Factor x^{2}+74x+1369. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+37\right)^{2}=8346$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+37\right)^{2}}=\sqrt{8346}$$

Simplify.

$$x+37=\sqrt{8346} x+37=-\sqrt{8346}$$

Subtract 37 from both sides of the equation.

$$x=\sqrt{8346}-37 x=-\sqrt{8346}-37$$