Question

select the correct answer from each drop - down menu. consider this equation ((4x)^{\frac{1}{2}}-x = 0) the first step in solving this equation is to (\boldsymbol{downarrow}) the second step is to (\boldsymbol{downarrow}) solving this equation for (x) initially yields (\boldsymbol{downarrow}) checking the solutions shows that (\boldsymbol{downarrow}) reset next

Explanation:

Step1: Isolate the radical term

To solve the equation \((4x)^{\frac{1}{2}} - x = 0\), the first step is to isolate the radical term. So we add \(x\) to both sides of the equation:

$$ (4x)^{\frac{1}{2}} = x $$

Step2: Eliminate the square root

The second step is to eliminate the square root by squaring both sides of the equation. Squaring the left - hand side \((4x)^{\frac{1}{2}}\) gives \(4x\), and squaring the right - hand side \(x\) gives \(x^{2}\). So we get:

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

Step3: Solve the quadratic equation

Rearrange the equation \(4x = x^{2}\) to the standard quadratic form \(x^{2}-4x = 0\). Factor out an \(x\):

$$ x(x - 4)=0 $$

Using the zero - product property, we have \(x = 0\) or \(x-4=0\), so \(x = 0\) or \(x = 4\).

Step4: Check the solutions

• For \(x = 0\): Substitute \(x = 0\) into the original equation \((4x)^{\frac{1}{2}}-x\). We get \((4\times0)^{\frac{1}{2}}-0=\sqrt{0}-0 = 0\), so \(x = 0\) is a valid solution.
• For \(x = 4\): Substitute \(x = 4\) into the original equation \((4x)^{\frac{1}{2}}-x\). We get \((4\times4)^{\frac{1}{2}}-4=\sqrt{16}-4 = 4 - 4=0\), so \(x = 4\) is a valid solution.

Answer:

First step: Isolate the radical term (add \(x\) to both sides, \((4x)^{\frac{1}{2}}=x\)); Second step: Square both sides (\(4x = x^{2}\)); Solutions: \(x = 0\) and \(x = 4\) (both are valid).