Solve 2x=sqrt{10+3x}

$$2 x = \sqrt { 10 + 3 x }$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(2x\right)^{2}=\left(\sqrt{10+3x}\right)^{2}$$

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

$$2^{2}x^{2}=\left(\sqrt{10+3x}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4x^{2}=\left(\sqrt{10+3x}\right)^{2}$$

Calculate \sqrt{10+3x} to the power of 2 and get 10+3x.

$$4x^{2}=10+3x$$

Subtract 10 from both sides.

$$4x^{2}-10=3x$$

Subtract 3x from both sides.

$$4x^{2}-10-3x=0$$

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

$$4x^{2}-3x-10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-10. To find a and b, set up a system to be solved.

$$a+b=-3 ab=4\left(-10\right)=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Calculate the sum for each pair.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

The solution is the pair that gives sum -3.

$$a=-8 b=5$$

Rewrite 4x^{2}-3x-10 as \left(4x^{2}-8x\right)+\left(5x-10\right).

$$\left(4x^{2}-8x\right)+\left(5x-10\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(x-2\right)+5\left(x-2\right)$$

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

$$\left(x-2\right)\left(4x+5\right)$$

To find equation solutions, solve x-2=0 and 4x+5=0.

$$x=2 x=-\frac{5}{4}$$

Substitute 2 for x in the equation 2x=\sqrt{10+3x}.

$$2\times 2=\sqrt{10+3\times 2}$$

Simplify. The value x=2 satisfies the equation.

$$4=4$$

Substitute -\frac{5}{4} for x in the equation 2x=\sqrt{10+3x}.

$$2\left(-\frac{5}{4}\right)=\sqrt{10+3\left(-\frac{5}{4}\right)}$$

Simplify. The value x=-\frac{5}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{5}{2}=\frac{5}{2}$$

Equation 2x=\sqrt{3x+10} has a unique solution.

$$x=2$$