Solve x^2+140x=261

$$x ^ { 2 } + 140 x = 261$$

Solve for x (complex solution)

$$x=\sqrt{5161}-70\approx 1.840100223$$

$$x=-\left(\sqrt{5161}+70\right)\approx -141.840100223$$

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.

$$x^{2}+140x=261$$

Subtract 261 from both sides of the equation.

$$x^{2}+140x-261=261-261$$

Subtracting 261 from itself leaves 0.

$$x^{2}+140x-261=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 140 for b, and -261 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-261\right)}}{2}$$

Square 140.

$$x=\frac{-140±\sqrt{19600-4\left(-261\right)}}{2}$$

Multiply -4 times -261.

$$x=\frac{-140±\sqrt{19600+1044}}{2}$$

Add 19600 to 1044.

$$x=\frac{-140±\sqrt{20644}}{2}$$

Take the square root of 20644.

$$x=\frac{-140±2\sqrt{5161}}{2}$$

Now solve the equation x=\frac{-140±2\sqrt{5161}}{2} when ± is plus. Add -140 to 2\sqrt{5161}.

$$x=\frac{2\sqrt{5161}-140}{2}$$

Divide -140+2\sqrt{5161} by 2.

$$x=\sqrt{5161}-70$$

Now solve the equation x=\frac{-140±2\sqrt{5161}}{2} when ± is minus. Subtract 2\sqrt{5161} from -140.

$$x=\frac{-2\sqrt{5161}-140}{2}$$

Divide -140-2\sqrt{5161} by 2.

$$x=-\sqrt{5161}-70$$

The equation is now solved.

$$x=\sqrt{5161}-70 x=-\sqrt{5161}-70$$

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.

$$x^{2}+140x=261$$

Divide 140, the coefficient of the x term, by 2 to get 70. Then add the square of 70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+140x+70^{2}=261+70^{2}$$

Square 70.

$$x^{2}+140x+4900=261+4900$$

Add 261 to 4900.

$$x^{2}+140x+4900=5161$$

Factor x^{2}+140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+70\right)^{2}=5161$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+70\right)^{2}}=\sqrt{5161}$$

Simplify.

$$x+70=\sqrt{5161} x+70=-\sqrt{5161}$$

Subtract 70 from both sides of the equation.

$$x=\sqrt{5161}-70 x=-\sqrt{5161}-70$$

Solve for x

$$x=\sqrt{5161}-70\approx 1.840100223$$

$$x=-\sqrt{5161}-70\approx -141.840100223$$

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.

$$x^{2}+140x=261$$

Subtract 261 from both sides of the equation.

$$x^{2}+140x-261=261-261$$

Subtracting 261 from itself leaves 0.

$$x^{2}+140x-261=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 140 for b, and -261 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-261\right)}}{2}$$

Square 140.

$$x=\frac{-140±\sqrt{19600-4\left(-261\right)}}{2}$$

Multiply -4 times -261.

$$x=\frac{-140±\sqrt{19600+1044}}{2}$$

Add 19600 to 1044.

$$x=\frac{-140±\sqrt{20644}}{2}$$

Take the square root of 20644.

$$x=\frac{-140±2\sqrt{5161}}{2}$$

Now solve the equation x=\frac{-140±2\sqrt{5161}}{2} when ± is plus. Add -140 to 2\sqrt{5161}.

$$x=\frac{2\sqrt{5161}-140}{2}$$

Divide -140+2\sqrt{5161} by 2.

$$x=\sqrt{5161}-70$$

Now solve the equation x=\frac{-140±2\sqrt{5161}}{2} when ± is minus. Subtract 2\sqrt{5161} from -140.

$$x=\frac{-2\sqrt{5161}-140}{2}$$

Divide -140-2\sqrt{5161} by 2.

$$x=-\sqrt{5161}-70$$

The equation is now solved.

$$x=\sqrt{5161}-70 x=-\sqrt{5161}-70$$

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.

$$x^{2}+140x=261$$

Divide 140, the coefficient of the x term, by 2 to get 70. Then add the square of 70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+140x+70^{2}=261+70^{2}$$

Square 70.

$$x^{2}+140x+4900=261+4900$$

Add 261 to 4900.

$$x^{2}+140x+4900=5161$$

Factor x^{2}+140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+70\right)^{2}=5161$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+70\right)^{2}}=\sqrt{5161}$$

Simplify.

$$x+70=\sqrt{5161} x+70=-\sqrt{5161}$$

Subtract 70 from both sides of the equation.

$$x=\sqrt{5161}-70 x=-\sqrt{5161}-70$$