Question

If $ a $ and $ b $ are the roots of the equation $ 4x^2 - 12x + 11 = 0 $, find the value of $ a^2 + b^2 $.

• A. \(\frac{7}{2}\)
• B. \(\frac{5}{2}\)
• C. \(\frac{3}{2}\)
• D. \(\frac{1}{2}\)

Hint:

To find \( a^2 + b^2 \) for the roots of a quadratic equation, use the identity \( a^2 + b^2 = (a + b)^2 - 2ab \) with the sum and product of roots.

Answer 1

The Correct Option is A

The given quadratic equation is \( 4x^2 - 12x + 11 = 0 \). For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum and product of the roots are: - Sum of roots: \( a + b = -\frac{b}{a} \) - Product of roots: \( a b = \frac{c}{a} \) Here, \( a = 4 \), \( b = -12 \), \( c = 11 \). Thus: \[ a + b = -\frac{-12}{4} = 3 \] \[ a b = \frac{11}{4} \] To find \( a^2 + b^2 \), use the identity: \[ a^2 + b^2 = (a + b)^2 - 2ab \] Substitute the values: \[ a^2 + b^2 = (3)^2 - 2 \cdot \frac{11}{4} = 9 - \frac{22}{4} = 9 - \frac{11}{2} = \frac{18 - 11}{2} = \frac{7}{2} \] Thus, the value of \( a^2 + b^2 \) is: \[ \boxed{\frac{7}{2}} \]