form a quadratic equation whose roots are the squares of the roots of 3x^2-5x-2=0

To form a quadratic equation whose roots are the squares of the roots of \( 3x^2 - 5x - 2 = 0 \), follow these steps:

### Step 1: Find the roots of the original quadratic equation
The given equation is:

\[
3x^2 - 5x - 2 = 0
\]

Using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 3 \), \( b = -5 \), and \( c = -2 \). Plugging in these values:

\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-2)}}{2(3)}
\]
\[
x = \frac{5 \pm \sqrt{25 + 24}}{6}
\]
\[
x = \frac{5 \pm \sqrt{49}}{6}
\]
\[
x = \frac{5 \pm 7}{6}
\]

Thus, the roots of the equation are:

\[
x_1 = \frac{5 + 7}{6} = 2 \quad \text{and} \quad x_2 = \frac{5 - 7}{6} = -\frac{1}{3}
\]

### Step 2: Square the roots
We are tasked with forming a new quadratic equation where the roots are the squares of the original roots. So, we square the roots \( x_1 \) and \( x_2 \):

\[
x_1^2 = 2^2 = 4 \quad \text{and} \quad x_2^2 = \left( -\frac{1}{3} \right)^2 = \frac{1}{9}
\]

### Step 3: Form the quadratic equation
The sum and product of the new roots (which are \( 4 \) and \( \frac{1}{9} \)) are:

- Sum of the roots: \( 4 + \frac{1}{9} = \frac{36}{9} + \frac{1}{9} = \frac{37}{9} \)
- Product of the roots: \( 4 \times \frac{1}{9} = \frac{4}{9} \)

Using the standard form of a quadratic equation \( x^2 - ( \text{sum of roots} ) x + \text{product of roots} = 0 \), we substitute the values:

\[
x^2 - \left( \frac{37}{9} \right) x + \frac{4}{9} = 0
\]

### Step 4: Clear the fractions
Multiply the entire equation by 9 to eliminate the fractions:

\[
9x^2 - 37x + 4 = 0
\]

Thus, the quadratic equation whose roots are the squares of the roots of \( 3x^2 - 5x - 2 = 0 \) is:

\[
9x^2 - 37x + 4 = 0
\]

Would you like further details or have any questions?

### Related Questions:
1. How can we solve a quadratic equation by factoring instead of using the quadratic formula?
2. What are the properties of the discriminant in a quadratic equation?
3. How do we derive the quadratic formula from the general form \( ax^2 + bx + c = 0 \)?
4. Can we form cubic equations from quadratic equations?
5. What are the geometric interpretations of quadratic equations?

### Tip:
Always double-check your calculations when using the quadratic formula, especially under the square root.

Form a quadratic equation whose roots are the squares of the roots of 3x^2 - 5x - 2 = 0.

Discover how to form a new quadratic equation where the roots are the squares of the roots of the equation 3x^2 - 5x - 2 = 0. This detailed guide involves using the quadratic formula and algebraic manipulation. Suitable for students in Grades 9-10.

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