Solve (sqrt{6x+7})-(2x-1)=0

$$( \sqrt { 6 x + 7 } ) - ( 2 x - 1 ) = 0$$

Solve for x

$$x=3$$

Solution Steps

Subtract -\left(2x-1\right) from both sides of the equation.

$$\sqrt{6x+7}=2x-1$$

Square both sides of the equation.

$$\left(\sqrt{6x+7}\right)^{2}=\left(2x-1\right)^{2}$$

Calculate \sqrt{6x+7} to the power of 2 and get 6x+7.

$$6x+7=\left(2x-1\right)^{2}$$

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

$$6x+7=4x^{2}-4x+1$$

Subtract 4x^{2} from both sides.

$$6x+7-4x^{2}=-4x+1$$

Add 4x to both sides.

$$6x+7-4x^{2}+4x=1$$

Combine 6x and 4x to get 10x.

$$10x+7-4x^{2}=1$$

Subtract 1 from both sides.

$$10x+7-4x^{2}-1=0$$

Subtract 1 from 7 to get 6.

$$10x+6-4x^{2}=0$$

Divide both sides by 2.

$$5x+3-2x^{2}=0$$

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

$$-2x^{2}+5x+3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+3. To find a and b, set up a system to be solved.

$$a+b=5 ab=-2\times 3=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 5.

$$a=6 b=-1$$

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

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

Factor out 2x in -2x^{2}+6x.

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

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(2x+1\right)$$

To find equation solutions, solve -x+3=0 and 2x+1=0.

$$x=3 x=-\frac{1}{2}$$

Substitute 3 for x in the equation \sqrt{6x+7}-\left(2x-1\right)=0.

$$\sqrt{6\times 3+7}-\left(2\times 3-1\right)=0$$

Simplify. The value x=3 satisfies the equation.

$$0=0$$

Substitute -\frac{1}{2} for x in the equation \sqrt{6x+7}-\left(2x-1\right)=0.

$$\sqrt{6\left(-\frac{1}{2}\right)+7}-\left(2\left(-\frac{1}{2}\right)-1\right)=0$$

Simplify. The value x=-\frac{1}{2} does not satisfy the equation.

$$4=0$$

Equation \sqrt{6x+7}=2x-1 has a unique solution.

$$x=3$$