Solve t=sqrt{t+2}+40

$$t = \sqrt { t + 2 } + 40$$

Solve for t

$$t=47$$

Solution Steps

Subtract 40 from both sides of the equation.

$$t-40=\sqrt{t+2}$$

Square both sides of the equation.

$$\left(t-40\right)^{2}=\left(\sqrt{t+2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(t-40\right)^{2}.

$$t^{2}-80t+1600=\left(\sqrt{t+2}\right)^{2}$$

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

$$t^{2}-80t+1600=t+2$$

Subtract t from both sides.

$$t^{2}-80t+1600-t=2$$

Combine -80t and -t to get -81t.

$$t^{2}-81t+1600=2$$

Subtract 2 from both sides.

$$t^{2}-81t+1600-2=0$$

Subtract 2 from 1600 to get 1598.

$$t^{2}-81t+1598=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -81 for b, and 1598 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 1598}}{2}$$

Square -81.

$$t=\frac{-\left(-81\right)±\sqrt{6561-4\times 1598}}{2}$$

Multiply -4 times 1598.

$$t=\frac{-\left(-81\right)±\sqrt{6561-6392}}{2}$$

Add 6561 to -6392.

$$t=\frac{-\left(-81\right)±\sqrt{169}}{2}$$

Take the square root of 169.

$$t=\frac{-\left(-81\right)±13}{2}$$

The opposite of -81 is 81.

$$t=\frac{81±13}{2}$$

Now solve the equation t=\frac{81±13}{2} when ± is plus. Add 81 to 13.

$$t=\frac{94}{2}$$

Divide 94 by 2.

$$t=47$$

Now solve the equation t=\frac{81±13}{2} when ± is minus. Subtract 13 from 81.

$$t=\frac{68}{2}$$

Divide 68 by 2.

$$t=34$$

The equation is now solved.

$$t=47 t=34$$

Substitute 47 for t in the equation t=\sqrt{t+2}+40.

$$47=\sqrt{47+2}+40$$

Simplify. The value t=47 satisfies the equation.

$$47=47$$

Substitute 34 for t in the equation t=\sqrt{t+2}+40.

$$34=\sqrt{34+2}+40$$

Simplify. The value t=34 does not satisfy the equation.

$$34=46$$

Equation t-40=\sqrt{t+2} has a unique solution.

$$t=47$$