Solve 2.56x^2=144+x^2-2*12x(-1/2)

$$2.56 { x }^{ 2 } = 144+ { x }^{ 2 } -2 \times 12x(- \frac{ 1 }{ 2 } )$$

Solve for x

$$x = \frac{10 \sqrt{181} + 50}{13} \approx 14.195095421$$

$$x=\frac{50-10\sqrt{181}}{13}\approx -6.502787729$$

Steps Using the Quadratic Formula

Multiply 2 and 12 to get 24.

$$2.56x^{2}=144+x^{2}-24x\left(-\frac{1}{2}\right)$$

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

$$2.56x^{2}=144+x^{2}-\left(-12x\right)$$

The opposite of -12x is 12x.

$$2.56x^{2}=144+x^{2}+12x$$

Subtract 144 from both sides.

$$2.56x^{2}-144=x^{2}+12x$$

Subtract x^{2} from both sides.

$$2.56x^{2}-144-x^{2}=12x$$

Combine 2.56x^{2} and -x^{2} to get 1.56x^{2}.

$$1.56x^{2}-144=12x$$

Subtract 12x from both sides.

$$1.56x^{2}-144-12x=0$$

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.

$$1.56x^{2}-12x-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.56 for a, -12 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 1.56\left(-144\right)}}{2\times 1.56}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 1.56\left(-144\right)}}{2\times 1.56}$$

Multiply -4 times 1.56.

$$x=\frac{-\left(-12\right)±\sqrt{144-6.24\left(-144\right)}}{2\times 1.56}$$

Multiply -6.24 times -144.

$$x=\frac{-\left(-12\right)±\sqrt{144+898.56}}{2\times 1.56}$$

Add 144 to 898.56.

$$x=\frac{-\left(-12\right)±\sqrt{1042.56}}{2\times 1.56}$$

Take the square root of 1042.56.

$$x=\frac{-\left(-12\right)±\frac{12\sqrt{181}}{5}}{2\times 1.56}$$

The opposite of -12 is 12.

$$x=\frac{12±\frac{12\sqrt{181}}{5}}{2\times 1.56}$$

Multiply 2 times 1.56.

$$x=\frac{12±\frac{12\sqrt{181}}{5}}{3.12}$$

Now solve the equation x=\frac{12±\frac{12\sqrt{181}}{5}}{3.12} when ± is plus. Add 12 to \frac{12\sqrt{181}}{5}.

$$x=\frac{\frac{12\sqrt{181}}{5}+12}{3.12}$$

Divide 12+\frac{12\sqrt{181}}{5} by 3.12 by multiplying 12+\frac{12\sqrt{181}}{5} by the reciprocal of 3.12.

$$x=\frac{10\sqrt{181}+50}{13}$$

Now solve the equation x=\frac{12±\frac{12\sqrt{181}}{5}}{3.12} when ± is minus. Subtract \frac{12\sqrt{181}}{5} from 12.

$$x=\frac{-\frac{12\sqrt{181}}{5}+12}{3.12}$$

Divide 12-\frac{12\sqrt{181}}{5} by 3.12 by multiplying 12-\frac{12\sqrt{181}}{5} by the reciprocal of 3.12.

$$x=\frac{50-10\sqrt{181}}{13}$$

The equation is now solved.

$$x=\frac{10\sqrt{181}+50}{13} x=\frac{50-10\sqrt{181}}{13}$$

Steps for Completing the Square

Multiply 2 and 12 to get 24.

$$2.56x^{2}=144+x^{2}-24x\left(-\frac{1}{2}\right)$$

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

$$2.56x^{2}=144+x^{2}-\left(-12x\right)$$

The opposite of -12x is 12x.

$$2.56x^{2}=144+x^{2}+12x$$

Subtract x^{2} from both sides.

$$2.56x^{2}-x^{2}=144+12x$$

Combine 2.56x^{2} and -x^{2} to get 1.56x^{2}.

$$1.56x^{2}=144+12x$$

Subtract 12x from both sides.

$$1.56x^{2}-12x=144$$

Divide both sides of the equation by 1.56, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.56x^{2}-12x}{1.56}=\frac{144}{1.56}$$

Dividing by 1.56 undoes the multiplication by 1.56.

$$x^{2}+\left(-\frac{12}{1.56}\right)x=\frac{144}{1.56}$$

Divide -12 by 1.56 by multiplying -12 by the reciprocal of 1.56.

$$x^{2}-\frac{100}{13}x=\frac{144}{1.56}$$

Divide 144 by 1.56 by multiplying 144 by the reciprocal of 1.56.

$$x^{2}-\frac{100}{13}x=\frac{1200}{13}$$

Divide -\frac{100}{13}, the coefficient of the x term, by 2 to get -\frac{50}{13}. Then add the square of -\frac{50}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{100}{13}x+\left(-\frac{50}{13}\right)^{2}=\frac{1200}{13}+\left(-\frac{50}{13}\right)^{2}$$

Square -\frac{50}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{100}{13}x+\frac{2500}{169}=\frac{1200}{13}+\frac{2500}{169}$$

Add \frac{1200}{13} to \frac{2500}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{100}{13}x+\frac{2500}{169}=\frac{18100}{169}$$

Factor x^{2}-\frac{100}{13}x+\frac{2500}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{50}{13}\right)^{2}=\frac{18100}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{50}{13}\right)^{2}}=\sqrt{\frac{18100}{169}}$$

Simplify.

$$x-\frac{50}{13}=\frac{10\sqrt{181}}{13} x-\frac{50}{13}=-\frac{10\sqrt{181}}{13}$$

Add \frac{50}{13} to both sides of the equation.

$$x=\frac{10\sqrt{181}+50}{13} x=\frac{50-10\sqrt{181}}{13}$$