The minimum value of the sum of the squares of the roots of x2 + (3 – a)x + 1 = 2a is
- 4
- 5
- 6
- 8
The Correct Option is C
Solution and Explanation
The correct answer is (C) : 6
α + β = a – 3, αβ = 1 – 2a
⇒ α2 + β2 = (a – 3)2 – 2(1– 2a)
= a2 – 6a + 9 – 2 + 4a
= a2 – 2a + 7
= (a – 1)2 + 6
So,
α2 + β2 ≥ 6
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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