Solve x+3=4sqrt{x}

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

Solve for x

$$x=9$$

$$x=1$$

Solution Steps

Square both sides of the equation.

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

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

$$x^{2}+6x+9=\left(4\sqrt{x}\right)^{2}$$

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

$$x^{2}+6x+9=4^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x^{2}+6x+9=16\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}+6x+9=16x$$

Subtract 16x from both sides.

$$x^{2}+6x+9-16x=0$$

Combine 6x and -16x to get -10x.

$$x^{2}-10x+9=0$$

To solve the equation, factor x^{2}-10x+9 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.

$$a+b=-10 ab=9$$

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 9.

$$-1,-9 -3,-3$$

Calculate the sum for each pair.

$$-1-9=-10 -3-3=-6$$

The solution is the pair that gives sum -10.

$$a=-9 b=-1$$

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

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

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

$$x=9 x=1$$

Substitute 9 for x in the equation x+3=4\sqrt{x}.

$$9+3=4\sqrt{9}$$

Simplify. The value x=9 satisfies the equation.

$$12=12$$

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

$$1+3=4\sqrt{1}$$

Simplify. The value x=1 satisfies the equation.

$$4=4$$

List all solutions of x+3=4\sqrt{x}.

$$x=9 x=1$$