Solve sqrt{7a-54}-a=-6

$$\sqrt { 7 a - 54 } - a = - 6$$

Solve for a

$$a=10$$

$$a=9$$

Solution Steps

Subtract -a from both sides of the equation.

$$\sqrt{7a-54}=-6+a$$

Square both sides of the equation.

$$\left(\sqrt{7a-54}\right)^{2}=\left(-6+a\right)^{2}$$

Calculate \sqrt{7a-54} to the power of 2 and get 7a-54.

$$7a-54=\left(-6+a\right)^{2}$$

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

$$7a-54=36-12a+a^{2}$$

Subtract 36 from both sides.

$$7a-54-36=-12a+a^{2}$$

Subtract 36 from -54 to get -90.

$$7a-90=-12a+a^{2}$$

Add 12a to both sides.

$$7a-90+12a=a^{2}$$

Combine 7a and 12a to get 19a.

$$19a-90=a^{2}$$

Subtract a^{2} from both sides.

$$19a-90-a^{2}=0$$

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

$$-a^{2}+19a-90=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -a^{2}+aa+ba-90. To find a and b, set up a system to be solved.

$$a+b=19 ab=-\left(-90\right)=90$$

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 90.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Calculate the sum for each pair.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

The solution is the pair that gives sum 19.

$$a=10 b=9$$

Rewrite -a^{2}+19a-90 as \left(-a^{2}+10a\right)+\left(9a-90\right).

$$\left(-a^{2}+10a\right)+\left(9a-90\right)$$

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

$$-a\left(a-10\right)+9\left(a-10\right)$$

Factor out common term a-10 by using distributive property.

$$\left(a-10\right)\left(-a+9\right)$$

To find equation solutions, solve a-10=0 and -a+9=0.

$$a=10 a=9$$

Substitute 10 for a in the equation \sqrt{7a-54}-a=-6.

$$\sqrt{7\times 10-54}-10=-6$$

Simplify. The value a=10 satisfies the equation.

$$-6=-6$$

Substitute 9 for a in the equation \sqrt{7a-54}-a=-6.

$$\sqrt{7\times 9-54}-9=-6$$

Simplify. The value a=9 satisfies the equation.

$$-6=-6$$

List all solutions of \sqrt{7a-54}=a-6.

$$a=10 a=9$$