Solve 2x+sqrt{x^2-9}=6

$$2 x + \sqrt { x ^ { 2 } - 9 } = 6$$

Solve for x

$$x=3$$

Solution Steps

Subtract 2x from both sides of the equation.

$$\sqrt{x^{2}-9}=6-2x$$

Square both sides of the equation.

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

Calculate \sqrt{x^{2}-9} to the power of 2 and get x^{2}-9.

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

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

$$x^{2}-9=36-24x+4x^{2}$$

Subtract 36 from both sides.

$$x^{2}-9-36=-24x+4x^{2}$$

Subtract 36 from -9 to get -45.

$$x^{2}-45=-24x+4x^{2}$$

Add 24x to both sides.

$$x^{2}-45+24x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$x^{2}-45+24x-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-45+24x=0$$

Divide both sides by 3.

$$-x^{2}-15+8x=0$$

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

$$-x^{2}+8x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=8 ab=-\left(-15\right)=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=5 b=3$$

Rewrite -x^{2}+8x-15 as \left(-x^{2}+5x\right)+\left(3x-15\right).

$$\left(-x^{2}+5x\right)+\left(3x-15\right)$$

Factor out -x in the first and 3 in the second group.

$$-x\left(x-5\right)+3\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+3\right)$$

To find equation solutions, solve x-5=0 and -x+3=0.

$$x=5 x=3$$

Substitute 5 for x in the equation 2x+\sqrt{x^{2}-9}=6.

$$2\times 5+\sqrt{5^{2}-9}=6$$

Simplify. The value x=5 does not satisfy the equation.

$$14=6$$

Substitute 3 for x in the equation 2x+\sqrt{x^{2}-9}=6.

$$2\times 3+\sqrt{3^{2}-9}=6$$

Simplify. The value x=3 satisfies the equation.

$$6=6$$

Equation \sqrt{x^{2}-9}=6-2x has a unique solution.

$$x=3$$