To solve the equation

$0 = \frac{-\frac{5x}{2} + \sqrt{x} + \frac{3}{2}}{\sqrt{\pi}},$

let's follow these steps:

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Step 1: Eliminate the denominator

Since $\sqrt{\pi} > 0$, multiplying through by $\sqrt{\pi}$ does not change the equation:

$0 = -\frac{5x}{2} + \sqrt{x} + \frac{3}{2}.$

This simplifies the problem to solving:

$-\frac{5x}{2} + \sqrt{x} + \frac{3}{2} = 0.$

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Step 2: Rearrange terms

Rearrange the terms to group the terms with $x$ on one side:

$\sqrt{x} = \frac{5x}{2} - \frac{3}{2}.$

Simplify further:

$\sqrt{x} = \frac{5x - 3}{2}.$

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Step 3: Square both sides

Squaring both sides of the equation removes the square root. This gives:

$x = \left(\frac{5x - 3}{2}\right)^2.$

Simplify the square:

$x = \frac{(5x - 3)^2}{4}.$

Multiply through by 4 to eliminate the fraction:

$4x = (5x - 3)^2.$

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Step 4: Expand the quadratic

Expand $(5x - 3)^2$:

$4x = 25x^2 - 30x + 9.$

Rearrange to form a standard quadratic equation:

$25x^2 - 34x + 9 = 0.$

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Step 5: Solve the quadratic equation

The quadratic equation is:

$25x^2 - 34x + 9 = 0.$

Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$

where $a = 25$, $b = -34$, and $c = 9$. Substitute these values:

$x = \frac{-(-34) \pm \sqrt{(-34)^2 - 4(25)(9)}}{2(25)}.$

Simplify:

$x = \frac{34 \pm \sqrt{1156 - 900}}{50}.$

$x = \frac{34 \pm \sqrt{256}}{50}.$

$x = \frac{34 \pm 16}{50}.$

This gives two solutions:

$x = \frac{34 + 16}{50} = \frac{50}{50} = 1,$

$x = \frac{34 - 16}{50} = \frac{18}{50} = \frac{9}{25}.$

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Step 6: Check for extraneous solutions

Both $x = 1$ and $x = \frac{9}{25}$ satisfy the original equation.

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Final Answer:

$x = 1 \quad \text{or} \quad x = \frac{9}{25}.$

Would you like me to verify the solutions or provide more explanation? 😊

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Related Questions:

1. How do you simplify square roots in equations like this?
2. What are extraneous solutions, and how do they arise?
3. Can you explain the quadratic formula in detail?
4. What is the significance of squaring both sides of an equation?
5. How does multiplying by the denominator affect the solution?

Tip:

Always check for extraneous solutions when squaring both sides of an equation!