Solve 3x+sqrt{5x+6}-x=0

$$3x+ \sqrt{ 5x+6 } -x=0$$

Solve for x

$$x=-\frac{3}{4}=-0.75$$

Solution Steps

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

$$\sqrt{5x+6}=-\left(3x-x\right)$$

Combine 3x and -x to get 2x.

$$\sqrt{5x+6}=-2x$$

Square both sides of the equation.

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

Calculate \sqrt{5x+6} to the power of 2 and get 5x+6.

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

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

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

Calculate -2 to the power of 2 and get 4.

$$5x+6=4x^{2}$$

Subtract 4x^{2} from both sides.

$$5x+6-4x^{2}=0$$

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

$$-4x^{2}+5x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=5 ab=-4\times 6=-24$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -24.

$$-1,24 -2,12 -3,8 -4,6$$

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 5.

$$a=8 b=-3$$

Rewrite -4x^{2}+5x+6 as \left(-4x^{2}+8x\right)+\left(-3x+6\right).

$$\left(-4x^{2}+8x\right)+\left(-3x+6\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(-x+2\right)+3\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(4x+3\right)$$

To find equation solutions, solve -x+2=0 and 4x+3=0.

$$x=2 x=-\frac{3}{4}$$

Substitute 2 for x in the equation 3x+\sqrt{5x+6}-x=0.

$$3\times 2+\sqrt{5\times 2+6}-2=0$$

Simplify. The value x=2 does not satisfy the equation.

$$8=0$$

Substitute -\frac{3}{4} for x in the equation 3x+\sqrt{5x+6}-x=0.

$$3\left(-\frac{3}{4}\right)+\sqrt{5\left(-\frac{3}{4}\right)+6}-\left(-\frac{3}{4}\right)=0$$

Simplify. The value x=-\frac{3}{4} satisfies the equation.

$$0=0$$

Equation \sqrt{5x+6}=-2x has a unique solution.

$$x=-\frac{3}{4}$$