Solve left(12-xright)^2+144=9x^2

$${ \left(12-x \right) }^{ 2 } +144=9 { x }^{ 2 }$$

Solve for x

$$x = \frac{3 \sqrt{17} - 3}{2} \approx 4.684658438$$

$$x=\frac{-3\sqrt{17}-3}{2}\approx -7.684658438$$

Steps Using the Quadratic Formula

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

$$144-24x+x^{2}+144=9x^{2}$$

Add 144 and 144 to get 288.

$$288-24x+x^{2}=9x^{2}$$

Subtract 9x^{2} from both sides.

$$288-24x+x^{2}-9x^{2}=0$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$288-24x-8x^{2}=0$$

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.

$$-8x^{2}-24x+288=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, -24 for b, and 288 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-\left(-24\right)±\sqrt{576+32\times 288}}{2\left(-8\right)}$$

Multiply 32 times 288.

$$x=\frac{-\left(-24\right)±\sqrt{576+9216}}{2\left(-8\right)}$$

Add 576 to 9216.

$$x=\frac{-\left(-24\right)±\sqrt{9792}}{2\left(-8\right)}$$

Take the square root of 9792.

$$x=\frac{-\left(-24\right)±24\sqrt{17}}{2\left(-8\right)}$$

The opposite of -24 is 24.

$$x=\frac{24±24\sqrt{17}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{24±24\sqrt{17}}{-16}$$

Now solve the equation x=\frac{24±24\sqrt{17}}{-16} when ± is plus. Add 24 to 24\sqrt{17}.

$$x=\frac{24\sqrt{17}+24}{-16}$$

Divide 24+24\sqrt{17} by -16.

$$x=\frac{-3\sqrt{17}-3}{2}$$

Now solve the equation x=\frac{24±24\sqrt{17}}{-16} when ± is minus. Subtract 24\sqrt{17} from 24.

$$x=\frac{24-24\sqrt{17}}{-16}$$

Divide 24-24\sqrt{17} by -16.

$$x=\frac{3\sqrt{17}-3}{2}$$

The equation is now solved.

$$x=\frac{-3\sqrt{17}-3}{2} x=\frac{3\sqrt{17}-3}{2}$$

Steps for Completing the Square

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

$$144-24x+x^{2}+144=9x^{2}$$

Add 144 and 144 to get 288.

$$288-24x+x^{2}=9x^{2}$$

Subtract 9x^{2} from both sides.

$$288-24x+x^{2}-9x^{2}=0$$

Combine x^{2} and -9x^{2} to get -8x^{2}.

$$288-24x-8x^{2}=0$$

Subtract 288 from both sides. Anything subtracted from zero gives its negation.

$$-24x-8x^{2}=-288$$

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.

$$-8x^{2}-24x=-288$$

Divide both sides by -8.

$$\frac{-8x^{2}-24x}{-8}=-\frac{288}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\left(-\frac{24}{-8}\right)x=-\frac{288}{-8}$$

Divide -24 by -8.

$$x^{2}+3x=-\frac{288}{-8}$$

Divide -288 by -8.

$$x^{2}+3x=36$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=36+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=36+\frac{9}{4}$$

Add 36 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{153}{4}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{153}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{153}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{3\sqrt{17}}{2} x+\frac{3}{2}=-\frac{3\sqrt{17}}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{3\sqrt{17}-3}{2} x=\frac{-3\sqrt{17}-3}{2}$$