Solve x^2+14x-720=1

$$x ^ { 2 } + 14 x - 720 = 1$$

Solve for x (complex solution)

$$x=\sqrt{770}-7\approx 20.748873851$$

$$x=-\left(\sqrt{770}+7\right)\approx -34.748873851$$

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.

$$x^{2}+14x-720=1$$

Subtract 1 from both sides of the equation.

$$x^{2}+14x-720-1=1-1$$

Subtracting 1 from itself leaves 0.

$$x^{2}+14x-720-1=0$$

Subtract 1 from -720.

$$x^{2}+14x-721=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -721 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-721\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-721\right)}}{2}$$

Multiply -4 times -721.

$$x=\frac{-14±\sqrt{196+2884}}{2}$$

Add 196 to 2884.

$$x=\frac{-14±\sqrt{3080}}{2}$$

Take the square root of 3080.

$$x=\frac{-14±2\sqrt{770}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{770}}{2} when ± is plus. Add -14 to 2\sqrt{770}.

$$x=\frac{2\sqrt{770}-14}{2}$$

Divide -14+2\sqrt{770} by 2.

$$x=\sqrt{770}-7$$

Now solve the equation x=\frac{-14±2\sqrt{770}}{2} when ± is minus. Subtract 2\sqrt{770} from -14.

$$x=\frac{-2\sqrt{770}-14}{2}$$

Divide -14-2\sqrt{770} by 2.

$$x=-\sqrt{770}-7$$

The equation is now solved.

$$x=\sqrt{770}-7 x=-\sqrt{770}-7$$

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.

$$x^{2}+14x-720=1$$

Add 720 to both sides of the equation.

$$x^{2}+14x-720-\left(-720\right)=1-\left(-720\right)$$

Subtracting -720 from itself leaves 0.

$$x^{2}+14x=1-\left(-720\right)$$

Subtract -720 from 1.

$$x^{2}+14x=721$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=721+7^{2}$$

Square 7.

$$x^{2}+14x+49=721+49$$

Add 721 to 49.

$$x^{2}+14x+49=770$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=770$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{770}$$

Simplify.

$$x+7=\sqrt{770} x+7=-\sqrt{770}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{770}-7 x=-\sqrt{770}-7$$

Solve for x

$$x=\sqrt{770}-7\approx 20.748873851$$

$$x=-\sqrt{770}-7\approx -34.748873851$$

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.

$$x^{2}+14x-720=1$$

Subtract 1 from both sides of the equation.

$$x^{2}+14x-720-1=1-1$$

Subtracting 1 from itself leaves 0.

$$x^{2}+14x-720-1=0$$

Subtract 1 from -720.

$$x^{2}+14x-721=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 14 for b, and -721 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-721\right)}}{2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\left(-721\right)}}{2}$$

Multiply -4 times -721.

$$x=\frac{-14±\sqrt{196+2884}}{2}$$

Add 196 to 2884.

$$x=\frac{-14±\sqrt{3080}}{2}$$

Take the square root of 3080.

$$x=\frac{-14±2\sqrt{770}}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{770}}{2} when ± is plus. Add -14 to 2\sqrt{770}.

$$x=\frac{2\sqrt{770}-14}{2}$$

Divide -14+2\sqrt{770} by 2.

$$x=\sqrt{770}-7$$

Now solve the equation x=\frac{-14±2\sqrt{770}}{2} when ± is minus. Subtract 2\sqrt{770} from -14.

$$x=\frac{-2\sqrt{770}-14}{2}$$

Divide -14-2\sqrt{770} by 2.

$$x=-\sqrt{770}-7$$

The equation is now solved.

$$x=\sqrt{770}-7 x=-\sqrt{770}-7$$

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.

$$x^{2}+14x-720=1$$

Add 720 to both sides of the equation.

$$x^{2}+14x-720-\left(-720\right)=1-\left(-720\right)$$

Subtracting -720 from itself leaves 0.

$$x^{2}+14x=1-\left(-720\right)$$

Subtract -720 from 1.

$$x^{2}+14x=721$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=721+7^{2}$$

Square 7.

$$x^{2}+14x+49=721+49$$

Add 721 to 49.

$$x^{2}+14x+49=770$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=770$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{770}$$

Simplify.

$$x+7=\sqrt{770} x+7=-\sqrt{770}$$

Subtract 7 from both sides of the equation.

$$x=\sqrt{770}-7 x=-\sqrt{770}-7$$