Solve x^2+144x+169=0

$${ x }^{ 2 } +144x+169=0$$

Solve for x (complex solution)

$$x=\sqrt{5015}-72\approx -1.183335295$$

$$x=-\left(\sqrt{5015}+72\right)\approx -142.816664705$$

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.

$$x^{2}+144x+169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 144 for b, and 169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-144±\sqrt{144^{2}-4\times 169}}{2}$$

Square 144.

$$x=\frac{-144±\sqrt{20736-4\times 169}}{2}$$

Multiply -4 times 169.

$$x=\frac{-144±\sqrt{20736-676}}{2}$$

Add 20736 to -676.

$$x=\frac{-144±\sqrt{20060}}{2}$$

Take the square root of 20060.

$$x=\frac{-144±2\sqrt{5015}}{2}$$

Now solve the equation x=\frac{-144±2\sqrt{5015}}{2} when ± is plus. Add -144 to 2\sqrt{5015}.

$$x=\frac{2\sqrt{5015}-144}{2}$$

Divide -144+2\sqrt{5015} by 2.

$$x=\sqrt{5015}-72$$

Now solve the equation x=\frac{-144±2\sqrt{5015}}{2} when ± is minus. Subtract 2\sqrt{5015} from -144.

$$x=\frac{-2\sqrt{5015}-144}{2}$$

Divide -144-2\sqrt{5015} by 2.

$$x=-\sqrt{5015}-72$$

The equation is now solved.

$$x=\sqrt{5015}-72 x=-\sqrt{5015}-72$$

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.

$$x^{2}+144x+169=0$$

Subtract 169 from both sides of the equation.

$$x^{2}+144x+169-169=-169$$

Subtracting 169 from itself leaves 0.

$$x^{2}+144x=-169$$

Divide 144, the coefficient of the x term, by 2 to get 72. Then add the square of 72 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+144x+72^{2}=-169+72^{2}$$

Square 72.

$$x^{2}+144x+5184=-169+5184$$

Add -169 to 5184.

$$x^{2}+144x+5184=5015$$

Factor x^{2}+144x+5184. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+72\right)^{2}=5015$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+72\right)^{2}}=\sqrt{5015}$$

Simplify.

$$x+72=\sqrt{5015} x+72=-\sqrt{5015}$$

Subtract 72 from both sides of the equation.

$$x=\sqrt{5015}-72 x=-\sqrt{5015}-72$$

Solve for x

$$x=\sqrt{5015}-72\approx -1.183335295$$

$$x=-\sqrt{5015}-72\approx -142.816664705$$

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.

$$x^{2}+144x+169=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 144 for b, and 169 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-144±\sqrt{144^{2}-4\times 169}}{2}$$

Square 144.

$$x=\frac{-144±\sqrt{20736-4\times 169}}{2}$$

Multiply -4 times 169.

$$x=\frac{-144±\sqrt{20736-676}}{2}$$

Add 20736 to -676.

$$x=\frac{-144±\sqrt{20060}}{2}$$

Take the square root of 20060.

$$x=\frac{-144±2\sqrt{5015}}{2}$$

Now solve the equation x=\frac{-144±2\sqrt{5015}}{2} when ± is plus. Add -144 to 2\sqrt{5015}.

$$x=\frac{2\sqrt{5015}-144}{2}$$

Divide -144+2\sqrt{5015} by 2.

$$x=\sqrt{5015}-72$$

Now solve the equation x=\frac{-144±2\sqrt{5015}}{2} when ± is minus. Subtract 2\sqrt{5015} from -144.

$$x=\frac{-2\sqrt{5015}-144}{2}$$

Divide -144-2\sqrt{5015} by 2.

$$x=-\sqrt{5015}-72$$

The equation is now solved.

$$x=\sqrt{5015}-72 x=-\sqrt{5015}-72$$

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.

$$x^{2}+144x+169=0$$

Subtract 169 from both sides of the equation.

$$x^{2}+144x+169-169=-169$$

Subtracting 169 from itself leaves 0.

$$x^{2}+144x=-169$$

Divide 144, the coefficient of the x term, by 2 to get 72. Then add the square of 72 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+144x+72^{2}=-169+72^{2}$$

Square 72.

$$x^{2}+144x+5184=-169+5184$$

Add -169 to 5184.

$$x^{2}+144x+5184=5015$$

Factor x^{2}+144x+5184. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+72\right)^{2}=5015$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+72\right)^{2}}=\sqrt{5015}$$

Simplify.

$$x+72=\sqrt{5015} x+72=-\sqrt{5015}$$

Subtract 72 from both sides of the equation.

$$x=\sqrt{5015}-72 x=-\sqrt{5015}-72$$