Solve 6/5*x+1=-4-sqrt{x+41}+5

$$\frac { 6 } { 5 } \cdot x + 1 = - 4 - \sqrt { x + 41 } + 5$$

Solve for x

$$x=-5$$

Solution Steps

Subtract -4+5 from both sides of the equation.

$$\frac{6}{5}x+1-\left(-4+5\right)=-\sqrt{x+41}$$

Add -4 and 5 to get 1.

$$\frac{6}{5}x+1-1=-\sqrt{x+41}$$

Subtract 1 from 1 to get 0.

$$\frac{6}{5}x=-\sqrt{x+41}$$

Square both sides of the equation.

$$\left(\frac{6}{5}x\right)^{2}=\left(-\sqrt{x+41}\right)^{2}$$

Expand \left(\frac{6}{5}x\right)^{2}.

$$\left(\frac{6}{5}\right)^{2}x^{2}=\left(-\sqrt{x+41}\right)^{2}$$

Calculate \frac{6}{5} to the power of 2 and get \frac{36}{25}.

$$\frac{36}{25}x^{2}=\left(-\sqrt{x+41}\right)^{2}$$

Expand \left(-\sqrt{x+41}\right)^{2}.

$$\frac{36}{25}x^{2}=\left(-1\right)^{2}\left(\sqrt{x+41}\right)^{2}$$

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

$$\frac{36}{25}x^{2}=1\left(\sqrt{x+41}\right)^{2}$$

Calculate \sqrt{x+41} to the power of 2 and get x+41.

$$\frac{36}{25}x^{2}=1\left(x+41\right)$$

Use the distributive property to multiply 1 by x+41.

$$\frac{36}{25}x^{2}=x+41$$

Subtract x from both sides.

$$\frac{36}{25}x^{2}-x=41$$

Subtract 41 from both sides.

$$\frac{36}{25}x^{2}-x-41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{36}{25} for a, -1 for b, and -41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times \frac{36}{25}\left(-41\right)}}{2\times \frac{36}{25}}$$

Multiply -4 times \frac{36}{25}.

$$x=\frac{-\left(-1\right)±\sqrt{1-\frac{144}{25}\left(-41\right)}}{2\times \frac{36}{25}}$$

Multiply -\frac{144}{25} times -41.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{5904}{25}}}{2\times \frac{36}{25}}$$

Add 1 to \frac{5904}{25}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{5929}{25}}}{2\times \frac{36}{25}}$$

Take the square root of \frac{5929}{25}.

$$x=\frac{-\left(-1\right)±\frac{77}{5}}{2\times \frac{36}{25}}$$

The opposite of -1 is 1.

$$x=\frac{1±\frac{77}{5}}{2\times \frac{36}{25}}$$

Multiply 2 times \frac{36}{25}.

$$x=\frac{1±\frac{77}{5}}{\frac{72}{25}}$$

Now solve the equation x=\frac{1±\frac{77}{5}}{\frac{72}{25}} when ± is plus. Add 1 to \frac{77}{5}.

$$x=\frac{\frac{82}{5}}{\frac{72}{25}}$$

Divide \frac{82}{5} by \frac{72}{25} by multiplying \frac{82}{5} by the reciprocal of \frac{72}{25}.

$$x=\frac{205}{36}$$

Now solve the equation x=\frac{1±\frac{77}{5}}{\frac{72}{25}} when ± is minus. Subtract \frac{77}{5} from 1.

$$x=-\frac{\frac{72}{5}}{\frac{72}{25}}$$

Divide -\frac{72}{5} by \frac{72}{25} by multiplying -\frac{72}{5} by the reciprocal of \frac{72}{25}.

$$x=-5$$

The equation is now solved.

$$x=\frac{205}{36} x=-5$$

Substitute \frac{205}{36} for x in the equation \frac{6}{5}x+1=-4-\sqrt{x+41}+5.

$$\frac{6}{5}\times \frac{205}{36}+1=-4-\sqrt{\frac{205}{36}+41}+5$$

Simplify. The value x=\frac{205}{36} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{47}{6}=-\frac{35}{6}$$

Substitute -5 for x in the equation \frac{6}{5}x+1=-4-\sqrt{x+41}+5.

$$\frac{6}{5}\left(-5\right)+1=-4-\sqrt{-5+41}+5$$

Simplify. The value x=-5 satisfies the equation.

$$-5=-5$$

Equation \frac{6x}{5}=-\sqrt{x+41} has a unique solution.

$$x=-5$$