The roots of the equation
$(3 -x)^4 + (2-x)^4$ = $(5 - 2x)^4$ are
- all real
- all imaginary
- two real and two imaginary
- none of these
The Correct Option is C
Solution and Explanation
Top Questions on Quadratic Equations
- If the roots of the quadratic equation \( x^2 - 7x + 12 = 0 \) are \( \alpha \) and \( \beta \), then the value of \( \alpha + \beta \) is:
- MHT CET - 2025
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For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:
\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \).
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let
\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).- GATE MA - 2025
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- If \( \alpha \) and \( \beta \) are negative real roots of the quadratic equation \( x^2 - (p + 2)x + (2p + 9) = 0 \) and \( p \in (\alpha, \beta) \). Then the value of \( \beta^2 - 2\alpha \) is:
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- The sum of the squares of all the roots of the equation \( x^2 + [2x - 3] - 4 = 0 \) is:
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- Let \( \alpha, \beta \) be the roots of the equation \( x^2 - ax - b = 0 \) with \( \text{Im}(\alpha) <\text{Im(\beta) \). Let \( P_n = \alpha^n - \beta^n \). If} \[ P_3 = -5\sqrt{7}, \, P_4 = -3\sqrt{7}, \, P_5 = 11\sqrt{7}, \, P_6 = 45\sqrt{7}, \] then \( |\alpha^4 + \beta^4| \) is equal to:
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Concepts Used:
Quadratic Equations
A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
The solution of a quadratic equation can be found using the formula, x=((-b±√(b²-4ac))/2a)
Two important points to keep in mind are:
- A polynomial equation has at least one root.
- A polynomial equation of degree ‘n’ has ‘n’ roots.
Read More: Nature of Roots of Quadratic Equation
There are basically four methods of solving quadratic equations. They are:
- Factoring
- Completing the square
- Using Quadratic Formula
- Taking the square root