Solve 5-x=sqrt{2x^2-71}

$$5 - x = \sqrt { 2 x ^ { 2 } - 71 }$$

Solve for x

$$x=-16$$

Solution Steps

Square both sides of the equation.

$$\left(5-x\right)^{2}=\left(\sqrt{2x^{2}-71}\right)^{2}$$

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

$$25-10x+x^{2}=\left(\sqrt{2x^{2}-71}\right)^{2}$$

Calculate \sqrt{2x^{2}-71} to the power of 2 and get 2x^{2}-71.

$$25-10x+x^{2}=2x^{2}-71$$

Subtract 2x^{2} from both sides.

$$25-10x+x^{2}-2x^{2}=-71$$

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

$$25-10x-x^{2}=-71$$

Add 71 to both sides.

$$25-10x-x^{2}+71=0$$

Add 25 and 71 to get 96.

$$96-10x-x^{2}=0$$

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

$$-x^{2}-10x+96=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+96. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-96=-96$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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.

$$1-96=-95 2-48=-46 3-32=-29 4-24=-20 6-16=-10 8-12=-4$$

The solution is the pair that gives sum -10.

$$a=6 b=-16$$

Rewrite -x^{2}-10x+96 as \left(-x^{2}+6x\right)+\left(-16x+96\right).

$$\left(-x^{2}+6x\right)+\left(-16x+96\right)$$

Factor out x in the first and 16 in the second group.

$$x\left(-x+6\right)+16\left(-x+6\right)$$

Factor out common term -x+6 by using distributive property.

$$\left(-x+6\right)\left(x+16\right)$$

To find equation solutions, solve -x+6=0 and x+16=0.

$$x=6 x=-16$$

Substitute 6 for x in the equation 5-x=\sqrt{2x^{2}-71}.

$$5-6=\sqrt{2\times 6^{2}-71}$$

Simplify. The value x=6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute -16 for x in the equation 5-x=\sqrt{2x^{2}-71}.

$$5-\left(-16\right)=\sqrt{2\left(-16\right)^{2}-71}$$

Simplify. The value x=-16 satisfies the equation.

$$21=21$$

Equation 5-x=\sqrt{2x^{2}-71} has a unique solution.

$$x=-16$$