Solve a-sqrt{a}=6

$$a- \sqrt{ a } =6$$

Solve for a

$$a=9$$

Solution Steps

Subtract a from both sides of the equation.

$$-\sqrt{a}=6-a$$

Square both sides of the equation.

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

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

$$\left(-1\right)^{2}\left(\sqrt{a}\right)^{2}=\left(6-a\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{a}\right)^{2}=\left(6-a\right)^{2}$$

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

$$1a=\left(6-a\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-a\right)^{2}.

$$1a=36-12a+a^{2}$$

Reorder the terms.

$$a=a^{2}-12a+36$$

Subtract a^{2} from both sides.

$$a-a^{2}=-12a+36$$

Add 12a to both sides.

$$a-a^{2}+12a=36$$

Combine a and 12a to get 13a.

$$13a-a^{2}=36$$

Subtract 36 from both sides.

$$13a-a^{2}-36=0$$

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

$$-a^{2}+13a-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -a^{2}+aa+ba-36. To find a and b, set up a system to be solved.

$$a+b=13 ab=-\left(-36\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 13.

$$a=9 b=4$$

Rewrite -a^{2}+13a-36 as \left(-a^{2}+9a\right)+\left(4a-36\right).

$$\left(-a^{2}+9a\right)+\left(4a-36\right)$$

Factor out -a in the first and 4 in the second group.

$$-a\left(a-9\right)+4\left(a-9\right)$$

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

$$\left(a-9\right)\left(-a+4\right)$$

To find equation solutions, solve a-9=0 and -a+4=0.

$$a=9 a=4$$

Substitute 9 for a in the equation a-\sqrt{a}=6.

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

Simplify. The value a=9 satisfies the equation.

$$6=6$$

Substitute 4 for a in the equation a-\sqrt{a}=6.

$$4-\sqrt{4}=6$$

Simplify. The value a=4 does not satisfy the equation.

$$2=6$$

Equation -\sqrt{a}=6-a has a unique solution.

$$a=9$$