Solve sqrt{16-t}-sqrt{3-t}=2

$$\sqrt { 16 - t } - \sqrt { 3 - t } = 2$$

Solve for t

$$t = -\frac{33}{16} = -2\frac{1}{16} = -2.0625$$

Solution Steps

Subtract -\sqrt{3-t} from both sides of the equation.

$$\sqrt{16-t}=2+\sqrt{3-t}$$

Square both sides of the equation.

$$\left(\sqrt{16-t}\right)^{2}=\left(2+\sqrt{3-t}\right)^{2}$$

Calculate \sqrt{16-t} to the power of 2 and get 16-t.

$$16-t=\left(2+\sqrt{3-t}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+\sqrt{3-t}\right)^{2}.

$$16-t=4+4\sqrt{3-t}+\left(\sqrt{3-t}\right)^{2}$$

Calculate \sqrt{3-t} to the power of 2 and get 3-t.

$$16-t=4+4\sqrt{3-t}+3-t$$

Add 4 and 3 to get 7.

$$16-t=7+4\sqrt{3-t}-t$$

Subtract 4\sqrt{3-t} from both sides.

$$16-t-4\sqrt{3-t}=7-t$$

Add t to both sides.

$$16-t-4\sqrt{3-t}+t=7$$

Combine -t and t to get 0.

$$16-4\sqrt{3-t}=7$$

Subtract 16 from both sides.

$$-4\sqrt{3-t}=7-16$$

Subtract 16 from 7 to get -9.

$$-4\sqrt{3-t}=-9$$

Divide both sides by -4.

$$\sqrt{3-t}=\frac{-9}{-4}$$

Fraction \frac{-9}{-4} can be simplified to \frac{9}{4} by removing the negative sign from both the numerator and the denominator.

$$\sqrt{3-t}=\frac{9}{4}$$

Square both sides of the equation.

$$-t+3=\frac{81}{16}$$

Subtract 3 from both sides of the equation.

$$-t+3-3=\frac{81}{16}-3$$

Subtracting 3 from itself leaves 0.

$$-t=\frac{81}{16}-3$$

Subtract 3 from \frac{81}{16}.

$$-t=\frac{33}{16}$$

Divide both sides by -1.

$$\frac{-t}{-1}=\frac{\frac{33}{16}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$t=\frac{\frac{33}{16}}{-1}$$

Divide \frac{33}{16} by -1.

$$t=-\frac{33}{16}$$

Substitute -\frac{33}{16} for t in the equation \sqrt{16-t}-\sqrt{3-t}=2.

$$\sqrt{16-\left(-\frac{33}{16}\right)}-\sqrt{3-\left(-\frac{33}{16}\right)}=2$$

Simplify. The value t=-\frac{33}{16} satisfies the equation.

$$2=2$$

Equation \sqrt{16-t}=\sqrt{3-t}+2 has a unique solution.

$$t=-\frac{33}{16}$$