Solve 2x-sqrt{x}-6=0

$$2 x - \sqrt { x } - 6 = 0$$

Solve for x

$$x=4$$

Solution Steps

Subtract 2x-6 from both sides of the equation.

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

Cancel out -1 on both sides.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-6\right)^{2}.

$$x=4x^{2}-24x+36$$

Subtract 4x^{2} from both sides.

$$x-4x^{2}=-24x+36$$

Add 24x to both sides.

$$x-4x^{2}+24x=36$$

Combine x and 24x to get 25x.

$$25x-4x^{2}=36$$

Subtract 36 from both sides.

$$25x-4x^{2}-36=0$$

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

$$-4x^{2}+25x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=25 ab=-4\left(-36\right)=144$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 144.

$$1,144 2,72 3,48 4,36 6,24 8,18 9,16 12,12$$

Calculate the sum for each pair.

$$1+144=145 2+72=74 3+48=51 4+36=40 6+24=30 8+18=26 9+16=25 12+12=24$$

The solution is the pair that gives sum 25.

$$a=16 b=9$$

Rewrite -4x^{2}+25x-36 as \left(-4x^{2}+16x\right)+\left(9x-36\right).

$$\left(-4x^{2}+16x\right)+\left(9x-36\right)$$

Factor out 4x in the first and -9 in the second group.

$$4x\left(-x+4\right)-9\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(4x-9\right)$$

To find equation solutions, solve -x+4=0 and 4x-9=0.

$$x=4 x=\frac{9}{4}$$

Substitute 4 for x in the equation 2x-\sqrt{x}-6=0.

$$2\times 4-\sqrt{4}-6=0$$

Simplify. The value x=4 satisfies the equation.

$$0=0$$

Substitute \frac{9}{4} for x in the equation 2x-\sqrt{x}-6=0.

$$2\times \frac{9}{4}-\sqrt{\frac{9}{4}}-6=0$$

Simplify. The value x=\frac{9}{4} does not satisfy the equation.

$$-3=0$$

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

$$x=4$$