Solve 2x-5=sqrt{x^2-7}

$$2 x - 5 = \sqrt { x ^ { 2 } - 7 }$$

Solve for x

$$x=4$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Solution Steps

Square both sides of the equation.

$$\left(2x-5\right)^{2}=\left(\sqrt{x^{2}-7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-5\right)^{2}.

$$4x^{2}-20x+25=\left(\sqrt{x^{2}-7}\right)^{2}$$

Calculate \sqrt{x^{2}-7} to the power of 2 and get x^{2}-7.

$$4x^{2}-20x+25=x^{2}-7$$

Subtract x^{2} from both sides.

$$4x^{2}-20x+25-x^{2}=-7$$

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

$$3x^{2}-20x+25=-7$$

Add 7 to both sides.

$$3x^{2}-20x+25+7=0$$

Add 25 and 7 to get 32.

$$3x^{2}-20x+32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+32. To find a and b, set up a system to be solved.

$$a+b=-20 ab=3\times 32=96$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 96.

$$-1,-96 -2,-48 -3,-32 -4,-24 -6,-16 -8,-12$$

Calculate the sum for each pair.

$$-1-96=-97 -2-48=-50 -3-32=-35 -4-24=-28 -6-16=-22 -8-12=-20$$

The solution is the pair that gives sum -20.

$$a=-12 b=-8$$

Rewrite 3x^{2}-20x+32 as \left(3x^{2}-12x\right)+\left(-8x+32\right).

$$\left(3x^{2}-12x\right)+\left(-8x+32\right)$$

Factor out 3x in the first and -8 in the second group.

$$3x\left(x-4\right)-8\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(3x-8\right)$$

To find equation solutions, solve x-4=0 and 3x-8=0.

$$x=4 x=\frac{8}{3}$$

Substitute 4 for x in the equation 2x-5=\sqrt{x^{2}-7}.

$$2\times 4-5=\sqrt{4^{2}-7}$$

Simplify. The value x=4 satisfies the equation.

$$3=3$$

Substitute \frac{8}{3} for x in the equation 2x-5=\sqrt{x^{2}-7}.

$$2\times \frac{8}{3}-5=\sqrt{\left(\frac{8}{3}\right)^{2}-7}$$

Simplify. The value x=\frac{8}{3} satisfies the equation.

$$\frac{1}{3}=\frac{1}{3}$$

List all solutions of 2x-5=\sqrt{x^{2}-7}.

$$x=4 x=\frac{8}{3}$$