Math Problem Statement

If α, β are roots of equation 2x^2 - 3x - 5 = 0, then find a quadratic equation whose roots are α^2, β^2.

Solution

The problem asks to find a quadratic equation whose roots are α2\alpha^2 and β2\beta^2, given that α\alpha and β\beta are the roots of the equation:

2x23x5=0.2x^2 - 3x - 5 = 0.

Step 1: Recall Vieta's formulas

For the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the sum and product of roots are: Sum of roots=ba,Product of roots=ca.\text{Sum of roots} = -\frac{b}{a}, \quad \text{Product of roots} = \frac{c}{a}. Here, a=2a = 2, b=3b = -3, and c=5c = -5. So: α+β=32=32,αβ=52.\alpha + \beta = -\frac{-3}{2} = \frac{3}{2}, \quad \alpha\beta = \frac{-5}{2}.

Step 2: Find the sum and product of α2\alpha^2 and β2\beta^2

The new roots are α2\alpha^2 and β2\beta^2. Using the formulas:

  • α2+β2=(α+β)22αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta,
  • α2β2=(αβ)2\alpha^2 \beta^2 = (\alpha \beta)^2.

Substitute the known values: α2+β2=(32)22(52)=94+5=294.\alpha^2 + \beta^2 = \left(\frac{3}{2}\right)^2 - 2\left(\frac{-5}{2}\right) = \frac{9}{4} + 5 = \frac{29}{4}. α2β2=(52)2=254.\alpha^2 \beta^2 = \left(\frac{-5}{2}\right)^2 = \frac{25}{4}.

Step 3: Form the quadratic equation

The quadratic equation with roots α2\alpha^2 and β2\beta^2 is: x2(α2+β2)x+α2β2=0.x^2 - (\alpha^2 + \beta^2)x + \alpha^2\beta^2 = 0. Substitute α2+β2=294\alpha^2 + \beta^2 = \frac{29}{4} and α2β2=254\alpha^2\beta^2 = \frac{25}{4}: x2294x+254=0.x^2 - \frac{29}{4}x + \frac{25}{4} = 0.

Multiply through by 4 to eliminate the fractions: 4x229x+25=0.4x^2 - 29x + 25 = 0.

Final Answer:

The quadratic equation is: 4x229x+25=0.4x^2 - 29x + 25 = 0.

Would you like further clarification or more examples? Here are 5 related questions you might find interesting:

  1. How do Vieta's formulas apply to higher-degree polynomials?
  2. Can we generalize this process for roots that are cubes (α3,β3\alpha^3, \beta^3)?
  3. How would the equation change if the roots were 1α\frac{1}{\alpha} and 1β\frac{1}{\beta}?
  4. What happens if α\alpha and β\beta are complex roots?
  5. How can we verify the solution graphically or numerically?

Tip: Always simplify expressions involving the sum and product of roots systematically to avoid algebraic errors!

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Roots and Coefficients
Sum and Product of Roots
Algebraic Manipulation

Formulas

Sum of roots = -(b/a)
Product of roots = c/a
Sum of squares of roots = (Sum of roots)^2 - 2(Product of roots)
Quadratic equation with roots α^2 and β^2: x^2 - (Sum of new roots)x + Product of new roots = 0

Theorems

Vieta's Formulas

Suitable Grade Level

Grades 10-12