Solve 2x=7sqrt{x}-3

$$2x = 7 \sqrt{ x } -3$$

Solve for x

$$x=9$$

$$x=\frac{1}{4}=0.25$$

Solution Steps

Subtract -3 from both sides of the equation.

$$2x+3=7\sqrt{x}$$

Square both sides of the equation.

$$\left(2x+3\right)^{2}=\left(7\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+3\right)^{2}.

$$4x^{2}+12x+9=\left(7\sqrt{x}\right)^{2}$$

Expand \left(7\sqrt{x}\right)^{2}.

$$4x^{2}+12x+9=7^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$4x^{2}+12x+9=49\left(\sqrt{x}\right)^{2}$$

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

$$4x^{2}+12x+9=49x$$

Subtract 49x from both sides.

$$4x^{2}+12x+9-49x=0$$

Combine 12x and -49x to get -37x.

$$4x^{2}-37x+9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+9. To find a and b, set up a system to be solved.

$$a+b=-37 ab=4\times 9=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Calculate the sum for each pair.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

The solution is the pair that gives sum -37.

$$a=-36 b=-1$$

Rewrite 4x^{2}-37x+9 as \left(4x^{2}-36x\right)+\left(-x+9\right).

$$\left(4x^{2}-36x\right)+\left(-x+9\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(x-9\right)-\left(x-9\right)$$

Factor out common term x-9 by using distributive property.

$$\left(x-9\right)\left(4x-1\right)$$

To find equation solutions, solve x-9=0 and 4x-1=0.

$$x=9 x=\frac{1}{4}$$

Substitute 9 for x in the equation 2x=7\sqrt{x}-3.

$$2\times 9=7\sqrt{9}-3$$

Simplify. The value x=9 satisfies the equation.

$$18=18$$

Substitute \frac{1}{4} for x in the equation 2x=7\sqrt{x}-3.

$$2\times \frac{1}{4}=7\sqrt{\frac{1}{4}}-3$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$\frac{1}{2}=\frac{1}{2}$$

List all solutions of 2x+3=7\sqrt{x}.

$$x=9 x=\frac{1}{4}$$