Solve sqrt{4-9x}=3x

$$\sqrt { 4 - 9 x } = 3 x$$

Solve for x

$$x=\frac{1}{3}\approx 0.333333333$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{4-9x}\right)^{2}=\left(3x\right)^{2}$$

Calculate \sqrt{4-9x} to the power of 2 and get 4-9x.

$$4-9x=\left(3x\right)^{2}$$

Expand \left(3x\right)^{2}.

$$4-9x=3^{2}x^{2}$$

Calculate 3 to the power of 2 and get 9.

$$4-9x=9x^{2}$$

Subtract 9x^{2} from both sides.

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

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

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

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

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

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 -36.

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

Calculate the sum for each pair.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

The solution is the pair that gives sum -9.

$$a=3 b=-12$$

Rewrite -9x^{2}-9x+4 as \left(-9x^{2}+3x\right)+\left(-12x+4\right).

$$\left(-9x^{2}+3x\right)+\left(-12x+4\right)$$

Factor out 3x in the first and 4 in the second group.

$$3x\left(-3x+1\right)+4\left(-3x+1\right)$$

Factor out common term -3x+1 by using distributive property.

$$\left(-3x+1\right)\left(3x+4\right)$$

To find equation solutions, solve -3x+1=0 and 3x+4=0.

$$x=\frac{1}{3} x=-\frac{4}{3}$$

Substitute \frac{1}{3} for x in the equation \sqrt{4-9x}=3x.

$$\sqrt{4-9\times \frac{1}{3}}=3\times \frac{1}{3}$$

Simplify. The value x=\frac{1}{3} satisfies the equation.

$$1=1$$

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

$$\sqrt{4-9\left(-\frac{4}{3}\right)}=3\left(-\frac{4}{3}\right)$$

Simplify. The value x=-\frac{4}{3} does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-4$$

Equation \sqrt{4-9x}=3x has a unique solution.

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