Solve x=sqrt{3x+4}

$$x = \sqrt { 3 x + 4 }$$

Solve for x

$$x=4$$

Solution Steps

Square both sides of the equation.

$$x^{2}=\left(\sqrt{3x+4}\right)^{2}$$

Calculate \sqrt{3x+4} to the power of 2 and get 3x+4.

$$x^{2}=3x+4$$

Subtract 3x from both sides.

$$x^{2}-3x=4$$

Subtract 4 from both sides.

$$x^{2}-3x-4=0$$

To solve the equation, factor x^{2}-3x-4 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-3 ab=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=-4 b=1$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-4\right)\left(x+1\right)$$

To find equation solutions, solve x-4=0 and x+1=0.

$$x=4 x=-1$$

Substitute 4 for x in the equation x=\sqrt{3x+4}.

$$4=\sqrt{3\times 4+4}$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

Substitute -1 for x in the equation x=\sqrt{3x+4}.

$$-1=\sqrt{3\left(-1\right)+4}$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x=\sqrt{3x+4} has a unique solution.

$$x=4$$