Solve sqrt{x^2+4}+sqrt{left(12-xright)^2+9}=13

$$\sqrt{ { x }^{ 2 } +4 } + \sqrt{ { \left(12-x \right) }^{ 2 } +9 } =13$$

Solve for x

$$x = \frac{24}{5} = 4\frac{4}{5} = 4.8$$

Solution Steps

Subtract \sqrt{\left(12-x\right)^{2}+9} from both sides of the equation.

$$\sqrt{x^{2}+4}=13-\sqrt{\left(12-x\right)^{2}+9}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(12-x\right)^{2}.

$$\sqrt{x^{2}+4}=13-\sqrt{144-24x+x^{2}+9}$$

Add 144 and 9 to get 153.

$$\sqrt{x^{2}+4}=13-\sqrt{153-24x+x^{2}}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}+4}\right)^{2}=\left(13-\sqrt{153-24x+x^{2}}\right)^{2}$$

Calculate \sqrt{x^{2}+4} to the power of 2 and get x^{2}+4.

$$x^{2}+4=\left(13-\sqrt{153-24x+x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-\sqrt{153-24x+x^{2}}\right)^{2}.

$$x^{2}+4=169-26\sqrt{153-24x+x^{2}}+\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Calculate \sqrt{153-24x+x^{2}} to the power of 2 and get 153-24x+x^{2}.

$$x^{2}+4=169-26\sqrt{153-24x+x^{2}}+153-24x+x^{2}$$

Add 169 and 153 to get 322.

$$x^{2}+4=322-26\sqrt{153-24x+x^{2}}-24x+x^{2}$$

Subtract 322-24x+x^{2} from both sides of the equation.

$$x^{2}+4-\left(322-24x+x^{2}\right)=-26\sqrt{153-24x+x^{2}}$$

To find the opposite of 322-24x+x^{2}, find the opposite of each term.

$$x^{2}+4-322+24x-x^{2}=-26\sqrt{153-24x+x^{2}}$$

Subtract 322 from 4 to get -318.

$$x^{2}-318+24x-x^{2}=-26\sqrt{153-24x+x^{2}}$$

Combine x^{2} and -x^{2} to get 0.

$$-318+24x=-26\sqrt{153-24x+x^{2}}$$

Square both sides of the equation.

$$\left(-318+24x\right)^{2}=\left(-26\sqrt{153-24x+x^{2}}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-318+24x\right)^{2}.

$$101124-15264x+576x^{2}=\left(-26\sqrt{153-24x+x^{2}}\right)^{2}$$

Expand \left(-26\sqrt{153-24x+x^{2}}\right)^{2}.

$$101124-15264x+576x^{2}=\left(-26\right)^{2}\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Calculate -26 to the power of 2 and get 676.

$$101124-15264x+576x^{2}=676\left(\sqrt{153-24x+x^{2}}\right)^{2}$$

Calculate \sqrt{153-24x+x^{2}} to the power of 2 and get 153-24x+x^{2}.

$$101124-15264x+576x^{2}=676\left(153-24x+x^{2}\right)$$

Use the distributive property to multiply 676 by 153-24x+x^{2}.

$$101124-15264x+576x^{2}=103428-16224x+676x^{2}$$

Subtract 103428 from both sides.

$$101124-15264x+576x^{2}-103428=-16224x+676x^{2}$$

Subtract 103428 from 101124 to get -2304.

$$-2304-15264x+576x^{2}=-16224x+676x^{2}$$

Add 16224x to both sides.

$$-2304-15264x+576x^{2}+16224x=676x^{2}$$

Combine -15264x and 16224x to get 960x.

$$-2304+960x+576x^{2}=676x^{2}$$

Subtract 676x^{2} from both sides.

$$-2304+960x+576x^{2}-676x^{2}=0$$

Combine 576x^{2} and -676x^{2} to get -100x^{2}.

$$-2304+960x-100x^{2}=0$$

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.

$$-100x^{2}+960x-2304=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -100 for a, 960 for b, and -2304 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-960±\sqrt{960^{2}-4\left(-100\right)\left(-2304\right)}}{2\left(-100\right)}$$

Square 960.

$$x=\frac{-960±\sqrt{921600-4\left(-100\right)\left(-2304\right)}}{2\left(-100\right)}$$

Multiply -4 times -100.

$$x=\frac{-960±\sqrt{921600+400\left(-2304\right)}}{2\left(-100\right)}$$

Multiply 400 times -2304.

$$x=\frac{-960±\sqrt{921600-921600}}{2\left(-100\right)}$$

Add 921600 to -921600.

$$x=\frac{-960±\sqrt{0}}{2\left(-100\right)}$$

Take the square root of 0.

$$x=-\frac{960}{2\left(-100\right)}$$

Multiply 2 times -100.

$$x=-\frac{960}{-200}$$

Reduce the fraction \frac{-960}{-200} to lowest terms by extracting and canceling out 40.

$$x=\frac{24}{5}$$

Substitute \frac{24}{5} for x in the equation \sqrt{x^{2}+4}+\sqrt{\left(12-x\right)^{2}+9}=13.

$$\sqrt{\left(\frac{24}{5}\right)^{2}+4}+\sqrt{\left(12-\frac{24}{5}\right)^{2}+9}=13$$

Simplify. The value x=\frac{24}{5} satisfies the equation.

$$13=13$$

Equation \sqrt{x^{2}+4}=-\sqrt{x^{2}-24x+153}+13 has a unique solution.

$$x=\frac{24}{5}$$