Question

If \(\alpha\) and \(\beta\) are the zeros of the polynomial \(2x^2 - 7x + 3\), then \(\alpha^2 + \beta^2\) is :

• A. \(\frac{37}{3}\)
• B. \(\frac{25}{16}\)
• C. \(\frac{37}{2}\)
• D. \(\frac{37}{4}\)

Hint:

For polynomial \(ax^2+bx+c\) with zeros \(\alpha, \beta\): 1. Sum of zeros: \(\alpha + \beta = -b/a\). 2. Product of zeros: \(\alpha\beta = c/a\). 3. Required: \(\alpha^2 + \beta^2\). Use identity: \(\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta\). Given \(2x^2 - 7x + 3\): \(a=2, b=-7, c=3\). \(\alpha + \beta = -(-7)/2 = 7/2\). \(\alpha\beta = 3/2\). \(\alpha^2 + \beta^2 = (7/2)^2 - 2(3/2) = \frac{49}{4} - 3 = \frac{49}{4} - \frac{12}{4} = \frac{37}{4}\).

Answer 1

The Correct Option is D

Concept: For a quadratic polynomial \(ax^2 + bx + c\), if \(\alpha\) and \(\beta\) are its zeros, then:
Sum of zeros: \(\alpha + \beta = -\frac{b}{a}\)
Product of zeros: \(\alpha\beta = \frac{c}{a}\) We also use the algebraic identity: \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\). Step 1: Identify coefficients a, b, and c from the given polynomial The given polynomial is \(2x^2 - 7x + 3\). Comparing with \(ax^2 + bx + c\):
\(a = 2\)
\(b = -7\)
\(c = 3\) Step 2: Calculate the sum of the zeros (\(\alpha + \beta\)) \[ \alpha + \beta = -\frac{b}{a} = -\frac{(-7)}{2} = \frac{7}{2} \] Step 3: Calculate the product of the zeros (\(\alpha\beta\)) \[ \alpha\beta = \frac{c}{a} = \frac{3}{2} \] Step 4: Use the identity to find \(\alpha^2 + \beta^2\) The identity is \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\). Substitute the values of \(\alpha + \beta\) and \(\alpha\beta\): \[ \alpha^2 + \beta^2 = \left(\frac{7}{2}\right)^2 - 2\left(\frac{3}{2}\right) \] \[ \alpha^2 + \beta^2 = \frac{7^2}{2^2} - \frac{2 \times 3}{2} \] \[ \alpha^2 + \beta^2 = \frac{49}{4} - 3 \] To subtract, find a common denominator (which is 4): \[ 3 = \frac{3 \times 4}{4} = \frac{12}{4} \] \[ \alpha^2 + \beta^2 = \frac{49}{4} - \frac{12}{4} \] \[ \alpha^2 + \beta^2 = \frac{49 - 12}{4} \] \[ \alpha^2 + \beta^2 = \frac{37}{4} \] The value of \(\alpha^2 + \beta^2\) is \(\frac{37}{4}\). This matches option (4).