If alpha and beta are the roots of the equation 2x2 - 4x + 7 = 0, find alpha2 + beta2.

March 28, 2026Mathematics

ScanSolve AI Answer

March 28, 2026

Answer

-3

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 4x + 7 = 0$ , find $\alpha^2 + \beta^2$ .

Step 1: Identify the coefficients. For the quadratic equation $ax^2 + bx + c = 0$ , we have $a = 2$ , $b = -4$ , $c = 7$ .

Step 2: Find the sum and product of the roots. Sum of roots: $\alpha + \beta = -\frac{b}{a}$ $\alpha + \beta = -\frac{-4}{2} = \frac{4}{2} = 2$ Product of roots: $\alpha \beta = \frac{c}{a}$ $\alpha \beta = \frac{7}{2}$

Step 3: Use the identity for $\alpha^2 + \beta^2$ . We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$ .

Step 4: Substitute the values of $\alpha + \beta$ and $\alpha \beta$ into the identity. $\alpha^2 + \beta^2 = (2)^2 - 2\left(\frac{7}{2}\right)$ $\alpha^2 + \beta^2 = 4 - 7$ $\alpha^2 + \beta^2 = -3$

The value of $\alpha^2 + \beta^2$ is $\boxed{-3}$ .

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3) If and are the roots of the equation 2x^2 - 4x + 7 = 0, find ^2 + ^2. Step 1: Identify the coefficients.

Asked on March 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 28, 2026

3) If and are the roots of the equation 2x^2 - 4x + 7 = 0, find ^2 + ^2. Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a = 2, b = -4, c = 7. Step 2: Find the sum and product of the roots. Sum of roots: + = -(b)/(a) + = -(-4)/(2) = (4)/(2) = 2 Product of roots: = (c)/(a) = (7)/(2) Step 3: Use the identity for ^2 + ^2. We know that ^2 + ^2 = ( + )^2 - 2 . Step 4: Substitute the values of + and into the identity. ^2 + ^2 = (2)^2 - 2((7)/(2)) ^2 + ^2 = 4 - 7 ^2 + ^2 = -3 The value of ^2 + ^2 is -3.

Asked on March 28, 2026Mathematics

ScanSolve AI AnswerAnswered on March 28, 2026

3) If and are the roots of the equation 2x^2 - 4x + 7 = 0, find ^2 + ^2. Step 1: Identify the coefficients. For the quadratic equation ax^2 + bx + c = 0, we have a = 2, b = -4, c = 7. Step 2: Find the sum and product of the roots. Sum of roots: + = -(b)/(a) + = -(-4)/(2) = (4)/(2) = 2 Product of roots: = (c)/(a) = (7)/(2) Step 3: Use the identity for ^2 + ^2. We know that ^2 + ^2 = ( + )^2 - 2 . Step 4: Substitute the values of + and into the identity. ^2 + ^2 = (2)^2 - 2((7)/(2)) ^2 + ^2 = 4 - 7 ^2 + ^2 = -3 The value of ^2 + ^2 is -3.