Question

Isolate one radical on one side of the equation.Raise each side of the equation to a power equal to the index of the radical and simplify. Check all proposed solutions in the original equation.

Answer 1

The given equation is

`\sqrt[]{3\text{ - 2x}}\text{ - 4x = 0}`

The first step is to add 4x to both sides of the equation. We have

`\begin{gathered} \sqrt[]{3\text{ - 2x}}\text{ - 4x + 4x = 0 + 4x} \\ \sqrt[]{3\text{ - 2x}}\text{ = 4x} \\ \text{Squaring both sides of the equation, we have} \\ (\sqrt[]{3-2x)}^2=(4x)^2 \\ 3-2x=16x^2 \end{gathered}`

3 - 2x = 16x^2

Adding 2x to both sides of the equation, we have

3 - 2x + 2x = 16x^2 + 2x

3 = 16x^2 + 2x

Subtracting 3 from both sides of the equation, we have

3 - 3 = 16x^2 + 2x - 3

0 = 16x^2 + 2x - 3

16x^2 + 2x - 3 = 0

This is a quadratic equation. We would solve for x by applying the method of factorisation. The first step is to multiply the first and last terms. We have 16x^2 * - 3 = - 48x^2. We would find two terms such that their sum or difference is 2x and their product is - 48x^2. The terms are 8x and - 6x. By replacing 2x with with 8x - 6x in the equation, we have

16x^2 + 8x - 6x - 3 = 0

By factorising, we have

8x(2x + 1) - 3(2x + 1) = 0

Since 2x + 1 is common, we have

(2x + 1)(8x - 3) = 0

2x + 1 = 0 or 8x - 3 = 0

2x = - 1 or 8x = 3

x = - 1/2 or x = 3/8

We would substitute these values in the original equation to check. We have

`\begin{gathered} For\text{ x = }-\text{ }(1)/(2) \\ \sqrt[]{3\text{ - 2}*-(1)/(2)}\text{ - 4}*-\text{ }(1)/(2)\text{ = 0} \\ \sqrt[]{3\text{ - - 1}}\text{ + 2 = 0} \\ \sqrt[]{4}\text{ + 2 = 0} \\ 2\text{ + 2 }\\e0 \end{gathered}`
`\begin{gathered} \text{For x = }(3)/(8) \\ \sqrt[]{3\text{ - 2}*(3)/(8)}\text{ - 4}*(3)/(8)\text{ = 0} \\ \sqrt[]{3\text{ - }(3)/(4)}\text{ - }(3)/(2)=\text{ 0} \\ \sqrt[]{(9)/(4)}\text{ - }(3)/(2)\text{ = 0} \\ (3)/(2)\text{ - }(3)/(2)\text{ = 0} \end{gathered}`

The solution is x = 3/8