Solve 4=sqrt{26+5x}+x

$$4 = \sqrt{ 26+5x } +x$$

Solve for x

$$x=\frac{13-\sqrt{209}}{2}\approx -0.728416147$$

Solution Steps

Subtract x from both sides of the equation.

$$4-x=\sqrt{26+5x}$$

Square both sides of the equation.

$$\left(4-x\right)^{2}=\left(\sqrt{26+5x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-x\right)^{2}.

$$16-8x+x^{2}=\left(\sqrt{26+5x}\right)^{2}$$

Calculate \sqrt{26+5x} to the power of 2 and get 26+5x.

$$16-8x+x^{2}=26+5x$$

Subtract 26 from both sides.

$$16-8x+x^{2}-26=5x$$

Subtract 26 from 16 to get -10.

$$-10-8x+x^{2}=5x$$

Subtract 5x from both sides.

$$-10-8x+x^{2}-5x=0$$

Combine -8x and -5x to get -13x.

$$-10-13x+x^{2}=0$$

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.

$$x^{2}-13x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -13 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-10\right)}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-10\right)}}{2}$$

Multiply -4 times -10.

$$x=\frac{-\left(-13\right)±\sqrt{169+40}}{2}$$

Add 169 to 40.

$$x=\frac{-\left(-13\right)±\sqrt{209}}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{209}}{2}$$

Now solve the equation x=\frac{13±\sqrt{209}}{2} when ± is plus. Add 13 to \sqrt{209}.

$$x=\frac{\sqrt{209}+13}{2}$$

Now solve the equation x=\frac{13±\sqrt{209}}{2} when ± is minus. Subtract \sqrt{209} from 13.

$$x=\frac{13-\sqrt{209}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{209}+13}{2} x=\frac{13-\sqrt{209}}{2}$$

Substitute \frac{\sqrt{209}+13}{2} for x in the equation 4=\sqrt{26+5x}+x.

$$4=\sqrt{26+5\times \frac{\sqrt{209}+13}{2}}+\frac{\sqrt{209}+13}{2}$$

Simplify. The value x=\frac{\sqrt{209}+13}{2} does not satisfy the equation.

$$4=9+209^{\frac{1}{2}}$$

Substitute \frac{13-\sqrt{209}}{2} for x in the equation 4=\sqrt{26+5x}+x.

$$4=\sqrt{26+5\times \frac{13-\sqrt{209}}{2}}+\frac{13-\sqrt{209}}{2}$$

Simplify. The value x=\frac{13-\sqrt{209}}{2} satisfies the equation.

$$4=4$$

Equation 4-x=\sqrt{5x+26} has a unique solution.

$$x=\frac{13-\sqrt{209}}{2}$$