Solve 256x^2=144+x^2-2*12x(-1/2)

$$256 { x }^{ 2 } = 144+ { x }^{ 2 } -2 \times 12x(- \frac{ 1 }{ 2 } )$$

Solve for x

$$x=\frac{2\sqrt{1021}+2}{85}\approx 0.775366838$$

$$x=\frac{2-2\sqrt{1021}}{85}\approx -0.728308015$$

Steps Using the Quadratic Formula

Multiply 2 and 12 to get 24.

$$256x^{2}=144+x^{2}-24x\left(-\frac{1}{2}\right)$$

Multiply 24 and -\frac{1}{2} to get -12.

$$256x^{2}=144+x^{2}-\left(-12x\right)$$

The opposite of -12x is 12x.

$$256x^{2}=144+x^{2}+12x$$

Subtract 144 from both sides.

$$256x^{2}-144=x^{2}+12x$$

Subtract x^{2} from both sides.

$$256x^{2}-144-x^{2}=12x$$

Combine 256x^{2} and -x^{2} to get 255x^{2}.

$$255x^{2}-144=12x$$

Subtract 12x from both sides.

$$255x^{2}-144-12x=0$$

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.

$$255x^{2}-12x-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 255 for a, -12 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 255\left(-144\right)}}{2\times 255}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 255\left(-144\right)}}{2\times 255}$$

Multiply -4 times 255.

$$x=\frac{-\left(-12\right)±\sqrt{144-1020\left(-144\right)}}{2\times 255}$$

Multiply -1020 times -144.

$$x=\frac{-\left(-12\right)±\sqrt{144+146880}}{2\times 255}$$

Add 144 to 146880.

$$x=\frac{-\left(-12\right)±\sqrt{147024}}{2\times 255}$$

Take the square root of 147024.

$$x=\frac{-\left(-12\right)±12\sqrt{1021}}{2\times 255}$$

The opposite of -12 is 12.

$$x=\frac{12±12\sqrt{1021}}{2\times 255}$$

Multiply 2 times 255.

$$x=\frac{12±12\sqrt{1021}}{510}$$

Now solve the equation x=\frac{12±12\sqrt{1021}}{510} when ± is plus. Add 12 to 12\sqrt{1021}.

$$x=\frac{12\sqrt{1021}+12}{510}$$

Divide 12+12\sqrt{1021} by 510.

$$x=\frac{2\sqrt{1021}+2}{85}$$

Now solve the equation x=\frac{12±12\sqrt{1021}}{510} when ± is minus. Subtract 12\sqrt{1021} from 12.

$$x=\frac{12-12\sqrt{1021}}{510}$$

Divide 12-12\sqrt{1021} by 510.

$$x=\frac{2-2\sqrt{1021}}{85}$$

The equation is now solved.

$$x=\frac{2\sqrt{1021}+2}{85} x=\frac{2-2\sqrt{1021}}{85}$$

Steps for Completing the Square

Multiply 2 and 12 to get 24.

$$256x^{2}=144+x^{2}-24x\left(-\frac{1}{2}\right)$$

Multiply 24 and -\frac{1}{2} to get -12.

$$256x^{2}=144+x^{2}-\left(-12x\right)$$

The opposite of -12x is 12x.

$$256x^{2}=144+x^{2}+12x$$

Subtract x^{2} from both sides.

$$256x^{2}-x^{2}=144+12x$$

Combine 256x^{2} and -x^{2} to get 255x^{2}.

$$255x^{2}=144+12x$$

Subtract 12x from both sides.

$$255x^{2}-12x=144$$

Divide both sides by 255.

$$\frac{255x^{2}-12x}{255}=\frac{144}{255}$$

Dividing by 255 undoes the multiplication by 255.

$$x^{2}+\left(-\frac{12}{255}\right)x=\frac{144}{255}$$

Reduce the fraction \frac{-12}{255} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{4}{85}x=\frac{144}{255}$$

Reduce the fraction \frac{144}{255} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{4}{85}x=\frac{48}{85}$$

Divide -\frac{4}{85}, the coefficient of the x term, by 2 to get -\frac{2}{85}. Then add the square of -\frac{2}{85} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{4}{85}x+\left(-\frac{2}{85}\right)^{2}=\frac{48}{85}+\left(-\frac{2}{85}\right)^{2}$$

Square -\frac{2}{85} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{4}{85}x+\frac{4}{7225}=\frac{48}{85}+\frac{4}{7225}$$

Add \frac{48}{85} to \frac{4}{7225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{4}{85}x+\frac{4}{7225}=\frac{4084}{7225}$$

Factor x^{2}-\frac{4}{85}x+\frac{4}{7225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2}{85}\right)^{2}=\frac{4084}{7225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2}{85}\right)^{2}}=\sqrt{\frac{4084}{7225}}$$

Simplify.

$$x-\frac{2}{85}=\frac{2\sqrt{1021}}{85} x-\frac{2}{85}=-\frac{2\sqrt{1021}}{85}$$

Add \frac{2}{85} to both sides of the equation.

$$x=\frac{2\sqrt{1021}+2}{85} x=\frac{2-2\sqrt{1021}}{85}$$