Solve sqrt{4x+1}-sqrt{9x-2}=+(-1)

$$\sqrt{ 4x+1 } - \sqrt{ 9x-2 } = +(-1)$$

Solve for x

$$x=2$$

Solution Steps

Subtract -\sqrt{9x-2} from both sides of the equation.

$$\sqrt{4x+1}=-1+\sqrt{9x-2}$$

Square both sides of the equation.

$$\left(\sqrt{4x+1}\right)^{2}=\left(-1+\sqrt{9x-2}\right)^{2}$$

Calculate \sqrt{4x+1} to the power of 2 and get 4x+1.

$$4x+1=\left(-1+\sqrt{9x-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-1+\sqrt{9x-2}\right)^{2}.

$$4x+1=1-2\sqrt{9x-2}+\left(\sqrt{9x-2}\right)^{2}$$

Calculate \sqrt{9x-2} to the power of 2 and get 9x-2.

$$4x+1=1-2\sqrt{9x-2}+9x-2$$

Subtract 2 from 1 to get -1.

$$4x+1=-1-2\sqrt{9x-2}+9x$$

Subtract -1+9x from both sides of the equation.

$$4x+1-\left(-1+9x\right)=-2\sqrt{9x-2}$$

To find the opposite of -1+9x, find the opposite of each term.

$$4x+1+1-9x=-2\sqrt{9x-2}$$

Add 1 and 1 to get 2.

$$4x+2-9x=-2\sqrt{9x-2}$$

Combine 4x and -9x to get -5x.

$$-5x+2=-2\sqrt{9x-2}$$

Square both sides of the equation.

$$\left(-5x+2\right)^{2}=\left(-2\sqrt{9x-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-5x+2\right)^{2}.

$$25x^{2}-20x+4=\left(-2\sqrt{9x-2}\right)^{2}$$

Expand \left(-2\sqrt{9x-2}\right)^{2}.

$$25x^{2}-20x+4=\left(-2\right)^{2}\left(\sqrt{9x-2}\right)^{2}$$

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

$$25x^{2}-20x+4=4\left(\sqrt{9x-2}\right)^{2}$$

Calculate \sqrt{9x-2} to the power of 2 and get 9x-2.

$$25x^{2}-20x+4=4\left(9x-2\right)$$

Use the distributive property to multiply 4 by 9x-2.

$$25x^{2}-20x+4=36x-8$$

Subtract 36x from both sides.

$$25x^{2}-20x+4-36x=-8$$

Combine -20x and -36x to get -56x.

$$25x^{2}-56x+4=-8$$

Add 8 to both sides.

$$25x^{2}-56x+4+8=0$$

Add 4 and 8 to get 12.

$$25x^{2}-56x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 25x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=-56 ab=25\times 12=300$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 300.

$$-1,-300 -2,-150 -3,-100 -4,-75 -5,-60 -6,-50 -10,-30 -12,-25 -15,-20$$

Calculate the sum for each pair.

$$-1-300=-301 -2-150=-152 -3-100=-103 -4-75=-79 -5-60=-65 -6-50=-56 -10-30=-40 -12-25=-37 -15-20=-35$$

The solution is the pair that gives sum -56.

$$a=-50 b=-6$$

Rewrite 25x^{2}-56x+12 as \left(25x^{2}-50x\right)+\left(-6x+12\right).

$$\left(25x^{2}-50x\right)+\left(-6x+12\right)$$

Factor out 25x in the first and -6 in the second group.

$$25x\left(x-2\right)-6\left(x-2\right)$$

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

$$\left(x-2\right)\left(25x-6\right)$$

To find equation solutions, solve x-2=0 and 25x-6=0.

$$x=2 x=\frac{6}{25}$$

Substitute 2 for x in the equation \sqrt{4x+1}-\sqrt{9x-2}=-1.

$$\sqrt{4\times 2+1}-\sqrt{9\times 2-2}=-1$$

Simplify. The value x=2 satisfies the equation.

$$-1=-1$$

Substitute \frac{6}{25} for x in the equation \sqrt{4x+1}-\sqrt{9x-2}=-1.

$$\sqrt{4\times \frac{6}{25}+1}-\sqrt{9\times \frac{6}{25}-2}=-1$$

Simplify. The value x=\frac{6}{25} does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Substitute 2 for x in the equation \sqrt{4x+1}-\sqrt{9x-2}=-1.

$$\sqrt{4\times 2+1}-\sqrt{9\times 2-2}=-1$$

Simplify. The value x=2 satisfies the equation.

$$-1=-1$$

Equation \sqrt{4x+1}=\sqrt{9x-2}-1 has a unique solution.

$$x=2$$