Solve sqrt{t}(sqrt{t}-6)=-9

$$\sqrt { t } ( \sqrt { t } - 6 ) = - 9$$

Solve for t

$$t=9$$

Solution Steps

Use the distributive property to multiply \sqrt{t} by \sqrt{t}-6.

$$\left(\sqrt{t}\right)^{2}-6\sqrt{t}=-9$$

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

$$t-6\sqrt{t}=-9$$

Subtract t from both sides of the equation.

$$-6\sqrt{t}=-9-t$$

Square both sides of the equation.

$$\left(-6\sqrt{t}\right)^{2}=\left(-9-t\right)^{2}$$

Expand \left(-6\sqrt{t}\right)^{2}.

$$\left(-6\right)^{2}\left(\sqrt{t}\right)^{2}=\left(-9-t\right)^{2}$$

Calculate -6 to the power of 2 and get 36.

$$36\left(\sqrt{t}\right)^{2}=\left(-9-t\right)^{2}$$

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

$$36t=\left(-9-t\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-9-t\right)^{2}.

$$36t=81+18t+t^{2}$$

Subtract 18t from both sides.

$$36t-18t=81+t^{2}$$

Combine 36t and -18t to get 18t.

$$18t=81+t^{2}$$

Subtract t^{2} from both sides.

$$18t-t^{2}=81$$

Subtract 81 from both sides.

$$18t-t^{2}-81=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-t^{2}+18t-81=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -t^{2}+at+bt-81. To find a and b, set up a system to be solved.

$$a+b=18 ab=-\left(-81\right)=81$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 81.

$$1,81 3,27 9,9$$

Calculate the sum for each pair.

$$1+81=82 3+27=30 9+9=18$$

The solution is the pair that gives sum 18.

$$a=9 b=9$$

Rewrite -t^{2}+18t-81 as \left(-t^{2}+9t\right)+\left(9t-81\right).

$$\left(-t^{2}+9t\right)+\left(9t-81\right)$$

Factor out -t in the first and 9 in the second group.

$$-t\left(t-9\right)+9\left(t-9\right)$$

Factor out common term t-9 by using distributive property.

$$\left(t-9\right)\left(-t+9\right)$$

To find equation solutions, solve t-9=0 and -t+9=0.

$$t=9 t=9$$

Substitute 9 for t in the equation \sqrt{t}\left(\sqrt{t}-6\right)=-9.

$$\sqrt{9}\left(\sqrt{9}-6\right)=-9$$

Simplify. The value t=9 satisfies the equation.

$$-9=-9$$

Substitute 9 for t in the equation \sqrt{t}\left(\sqrt{t}-6\right)=-9.

$$\sqrt{9}\left(\sqrt{9}-6\right)=-9$$

Simplify. The value t=9 satisfies the equation.

$$-9=-9$$

List all solutions of -6\sqrt{t}=-t-9.

$$t=9 t=9$$