Solve sqrt{x+94}=x+12

$$\sqrt { x + 94 } = x + 12$$

Solve for x

$$x=\frac{\sqrt{329}-23}{2}\approx -2.430821426$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+94}\right)^{2}=\left(x+12\right)^{2}$$

Calculate \sqrt{x+94} to the power of 2 and get x+94.

$$x+94=\left(x+12\right)^{2}$$

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

$$x+94=x^{2}+24x+144$$

Subtract x^{2} from both sides.

$$x+94-x^{2}=24x+144$$

Subtract 24x from both sides.

$$x+94-x^{2}-24x=144$$

Combine x and -24x to get -23x.

$$-23x+94-x^{2}=144$$

Subtract 144 from both sides.

$$-23x+94-x^{2}-144=0$$

Subtract 144 from 94 to get -50.

$$-23x-50-x^{2}=0$$

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.

$$-x^{2}-23x-50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -23 for b, and -50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\left(-1\right)\left(-50\right)}}{2\left(-1\right)}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\left(-1\right)\left(-50\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-23\right)±\sqrt{529+4\left(-50\right)}}{2\left(-1\right)}$$

Multiply 4 times -50.

$$x=\frac{-\left(-23\right)±\sqrt{529-200}}{2\left(-1\right)}$$

Add 529 to -200.

$$x=\frac{-\left(-23\right)±\sqrt{329}}{2\left(-1\right)}$$

The opposite of -23 is 23.

$$x=\frac{23±\sqrt{329}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{23±\sqrt{329}}{-2}$$

Now solve the equation x=\frac{23±\sqrt{329}}{-2} when ± is plus. Add 23 to \sqrt{329}.

$$x=\frac{\sqrt{329}+23}{-2}$$

Divide 23+\sqrt{329} by -2.

$$x=\frac{-\sqrt{329}-23}{2}$$

Now solve the equation x=\frac{23±\sqrt{329}}{-2} when ± is minus. Subtract \sqrt{329} from 23.

$$x=\frac{23-\sqrt{329}}{-2}$$

Divide 23-\sqrt{329} by -2.

$$x=\frac{\sqrt{329}-23}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{329}-23}{2} x=\frac{\sqrt{329}-23}{2}$$

Substitute \frac{-\sqrt{329}-23}{2} for x in the equation \sqrt{x+94}=x+12.

$$\sqrt{\frac{-\sqrt{329}-23}{2}+94}=\frac{-\sqrt{329}-23}{2}+12$$

Simplify. The value x=\frac{-\sqrt{329}-23}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 329^{\frac{1}{2}}\right)=-\frac{1}{2}\times 329^{\frac{1}{2}}+\frac{1}{2}$$

Substitute \frac{\sqrt{329}-23}{2} for x in the equation \sqrt{x+94}=x+12.

$$\sqrt{\frac{\sqrt{329}-23}{2}+94}=\frac{\sqrt{329}-23}{2}+12$$

Simplify. The value x=\frac{\sqrt{329}-23}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}\times 329^{\frac{1}{2}}=\frac{1}{2}\times 329^{\frac{1}{2}}+\frac{1}{2}$$

Equation \sqrt{x+94}=x+12 has a unique solution.

$$x=\frac{\sqrt{329}-23}{2}$$