Solve sqrt{7x+106}=2x+26

$$\sqrt { 7 x + 106 } = 2 x + 26$$

Solve for x

$$x=-10$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{7x+106}\right)^{2}=\left(2x+26\right)^{2}$$

Calculate \sqrt{7x+106} to the power of 2 and get 7x+106.

$$7x+106=\left(2x+26\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+26\right)^{2}.

$$7x+106=4x^{2}+104x+676$$

Subtract 4x^{2} from both sides.

$$7x+106-4x^{2}=104x+676$$

Subtract 104x from both sides.

$$7x+106-4x^{2}-104x=676$$

Combine 7x and -104x to get -97x.

$$-97x+106-4x^{2}=676$$

Subtract 676 from both sides.

$$-97x+106-4x^{2}-676=0$$

Subtract 676 from 106 to get -570.

$$-97x-570-4x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-4x^{2}-97x-570=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -97 for b, and -570 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-97\right)±\sqrt{\left(-97\right)^{2}-4\left(-4\right)\left(-570\right)}}{2\left(-4\right)}$$

Square -97.

$$x=\frac{-\left(-97\right)±\sqrt{9409-4\left(-4\right)\left(-570\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-97\right)±\sqrt{9409+16\left(-570\right)}}{2\left(-4\right)}$$

Multiply 16 times -570.

$$x=\frac{-\left(-97\right)±\sqrt{9409-9120}}{2\left(-4\right)}$$

Add 9409 to -9120.

$$x=\frac{-\left(-97\right)±\sqrt{289}}{2\left(-4\right)}$$

Take the square root of 289.

$$x=\frac{-\left(-97\right)±17}{2\left(-4\right)}$$

The opposite of -97 is 97.

$$x=\frac{97±17}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{97±17}{-8}$$

Now solve the equation x=\frac{97±17}{-8} when ± is plus. Add 97 to 17.

$$x=\frac{114}{-8}$$

Reduce the fraction \frac{114}{-8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{57}{4}$$

Now solve the equation x=\frac{97±17}{-8} when ± is minus. Subtract 17 from 97.

$$x=\frac{80}{-8}$$

Divide 80 by -8.

$$x=-10$$

The equation is now solved.

$$x=-\frac{57}{4} x=-10$$

Substitute -\frac{57}{4} for x in the equation \sqrt{7x+106}=2x+26.

$$\sqrt{7\left(-\frac{57}{4}\right)+106}=2\left(-\frac{57}{4}\right)+26$$

Simplify. The value x=-\frac{57}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{5}{2}=-\frac{5}{2}$$

Substitute -10 for x in the equation \sqrt{7x+106}=2x+26.

$$\sqrt{7\left(-10\right)+106}=2\left(-10\right)+26$$

Simplify. The value x=-10 satisfies the equation.

$$6=6$$

Equation \sqrt{7x+106}=2x+26 has a unique solution.

$$x=-10$$