Answer :

Let's solve each of these questions step by step.

  1. Solve for x in the equation: [tex]3x^2 - 4x = 0[/tex]

    We can solve this quadratic equation by factoring. First, factor out the greatest common factor, which is [tex]x[/tex]:

    [tex]x(3x - 4) = 0[/tex]

    Setting each factor equal to zero gives us the possible solutions:

    [tex]x = 0[/tex]
    [tex]3x - 4 = 0[/tex]

    Solving [tex]3x - 4 = 0[/tex]:

    [tex]3x = 4[/tex]
    [tex]x = \frac{4}{3}[/tex]

    Therefore, the solutions are [tex]x = 0[/tex] and [tex]x = \frac{4}{3}[/tex].

  2. Solve for x in the equation: [tex]x - 6 + \frac{2}{x} = 0[/tex], [tex]x \neq 0[/tex]

    First, move [tex]6[/tex] to the other side:

    [tex]x + \frac{2}{x} = 6[/tex]

    Multiply every term by [tex]x[/tex] to eliminate the fraction:

    [tex]x^2 + 2 = 6x[/tex]

    Rearrange it into a standard quadratic form:

    [tex]x^2 - 6x + 2 = 0[/tex]

    Solve this using the quadratic formula [tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex], where [tex]a = 1[/tex], [tex]b = -6[/tex], and [tex]c = 2[/tex]:

    [tex]x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}[/tex]
    [tex]x = \frac{6 \pm \sqrt{36 - 8}}{2}[/tex]
    [tex]x = \frac{6 \pm \sqrt{28}}{2}[/tex]
    [tex]x = \frac{6 \pm 2\sqrt{7}}{2}[/tex]
    [tex]x = 3 \pm \sqrt{7}[/tex]

    Approximating these solutions to two decimal places:

    [tex]x \approx 3 + 2.65 = 5.65[/tex]
    [tex]x \approx 3 - 2.65 = 0.35[/tex]

    Therefore, [tex]x \approx 5.65[/tex] and [tex]x \approx 0.35[/tex].

  3. Solve for x in the equation: [tex]x^{2/3} = 4[/tex]

    To solve for [tex]x[/tex], raise both sides to the power of [tex]\frac{3}{2}[/tex] to cancel the [tex]\frac{2}{3}[/tex] exponent:

    [tex]\left(x^{2/3}\right)^{3/2} = 4^{3/2}[/tex]

    Simplifying the left side gives:

    [tex]x = 4^{3/2}[/tex]

    Calculate [tex]4^{3/2}[/tex]. First calculate [tex]\sqrt{4}[/tex] which equals [tex]2[/tex], and then raise it to the power of [tex]3[/tex]:

    [tex]x = 2^3 = 8[/tex]

    Therefore, the solution is [tex]x = 8[/tex].

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