Solve 3x-1-sqrt{x+5}=2

$$3 x - 1 - \sqrt { x + 5 } = 2$$

Solve for x

$$x = \frac{\sqrt{217} + 19}{18} \approx 1.873939992$$

Solution Steps

Subtract 3x-1 from both sides of the equation.

$$-\sqrt{x+5}=2-\left(3x-1\right)$$

To find the opposite of 3x-1, find the opposite of each term.

$$-\sqrt{x+5}=2-3x-\left(-1\right)$$

The opposite of -1 is 1.

$$-\sqrt{x+5}=2-3x+1$$

Add 2 and 1 to get 3.

$$-\sqrt{x+5}=3-3x$$

Square both sides of the equation.

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

Expand \left(-\sqrt{x+5}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{x+5}\right)^{2}=\left(3-3x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{x+5}\right)^{2}=\left(3-3x\right)^{2}$$

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

$$1\left(x+5\right)=\left(3-3x\right)^{2}$$

Use the distributive property to multiply 1 by x+5.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-3x\right)^{2}.

$$x+5=9-18x+9x^{2}$$

Subtract 9 from both sides.

$$x+5-9=-18x+9x^{2}$$

Subtract 9 from 5 to get -4.

$$x-4=-18x+9x^{2}$$

Add 18x to both sides.

$$x-4+18x=9x^{2}$$

Combine x and 18x to get 19x.

$$19x-4=9x^{2}$$

Subtract 9x^{2} from both sides.

$$19x-4-9x^{2}=0$$

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.

$$-9x^{2}+19x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 19 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-9\right)\left(-4\right)}}{2\left(-9\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-9\right)\left(-4\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-19±\sqrt{361+36\left(-4\right)}}{2\left(-9\right)}$$

Multiply 36 times -4.

$$x=\frac{-19±\sqrt{361-144}}{2\left(-9\right)}$$

Add 361 to -144.

$$x=\frac{-19±\sqrt{217}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-19±\sqrt{217}}{-18}$$

Now solve the equation x=\frac{-19±\sqrt{217}}{-18} when ± is plus. Add -19 to \sqrt{217}.

$$x=\frac{\sqrt{217}-19}{-18}$$

Divide -19+\sqrt{217} by -18.

$$x=\frac{19-\sqrt{217}}{18}$$

Now solve the equation x=\frac{-19±\sqrt{217}}{-18} when ± is minus. Subtract \sqrt{217} from -19.

$$x=\frac{-\sqrt{217}-19}{-18}$$

Divide -19-\sqrt{217} by -18.

$$x=\frac{\sqrt{217}+19}{18}$$

The equation is now solved.

$$x=\frac{19-\sqrt{217}}{18} x=\frac{\sqrt{217}+19}{18}$$

Substitute \frac{19-\sqrt{217}}{18} for x in the equation 3x-1-\sqrt{x+5}=2.

$$3\times \frac{19-\sqrt{217}}{18}-1-\sqrt{\frac{19-\sqrt{217}}{18}+5}=2$$

Simplify. The value x=\frac{19-\sqrt{217}}{18} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{7}{3}-\frac{1}{3}\times 217^{\frac{1}{2}}=2$$

Substitute \frac{\sqrt{217}+19}{18} for x in the equation 3x-1-\sqrt{x+5}=2.

$$3\times \frac{\sqrt{217}+19}{18}-1-\sqrt{\frac{\sqrt{217}+19}{18}+5}=2$$

Simplify. The value x=\frac{\sqrt{217}+19}{18} satisfies the equation.

$$2=2$$

Equation -\sqrt{x+5}=3-3x has a unique solution.

$$x=\frac{\sqrt{217}+19}{18}$$