Solve 4x=sqrt{30+4x}

$$4 x = \sqrt { 30 + 4 x }$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Solution Steps

Square both sides of the equation.

$$\left(4x\right)^{2}=\left(\sqrt{30+4x}\right)^{2}$$

Expand \left(4x\right)^{2}.

$$4^{2}x^{2}=\left(\sqrt{30+4x}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}=\left(\sqrt{30+4x}\right)^{2}$$

Calculate \sqrt{30+4x} to the power of 2 and get 30+4x.

$$16x^{2}=30+4x$$

Subtract 30 from both sides.

$$16x^{2}-30=4x$$

Subtract 4x from both sides.

$$16x^{2}-30-4x=0$$

Divide both sides by 2.

$$8x^{2}-15-2x=0$$

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

$$8x^{2}-2x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-2 ab=8\left(-15\right)=-120$$

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 -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Calculate the sum for each pair.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

The solution is the pair that gives sum -2.

$$a=-12 b=10$$

Rewrite 8x^{2}-2x-15 as \left(8x^{2}-12x\right)+\left(10x-15\right).

$$\left(8x^{2}-12x\right)+\left(10x-15\right)$$

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

$$4x\left(2x-3\right)+5\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(4x+5\right)$$

To find equation solutions, solve 2x-3=0 and 4x+5=0.

$$x=\frac{3}{2} x=-\frac{5}{4}$$

Substitute \frac{3}{2} for x in the equation 4x=\sqrt{30+4x}.

$$4\times \frac{3}{2}=\sqrt{30+4\times \frac{3}{2}}$$

Simplify. The value x=\frac{3}{2} satisfies the equation.

$$6=6$$

Substitute -\frac{5}{4} for x in the equation 4x=\sqrt{30+4x}.

$$4\left(-\frac{5}{4}\right)=\sqrt{30+4\left(-\frac{5}{4}\right)}$$

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

$$-5=5$$

Equation 4x=\sqrt{4x+30} has a unique solution.

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