Let a and b are roots of \(x^2 – 7x – 1 = 0\). The value of \(\frac{(a_{21} + b_{21} + a_{17} + b_{17})}{(a_{19} + b_{19})}\) is?
29
49
53
51
The Correct Option is D
Approach Solution - 1
The correct answer is option (D) : 51
\(\frac{a^{17}(a^4+1)+b^{17}(b^4+1)}{a^{19}+b^{19}}\)
\(\alpha^2-1=7\alpha\)
\(\Rightarrow \alpha^4+1=51\alpha^2\)\(\large<^{\large{a}}_{\large{b}}\)
\(\therefore \frac{51a^{19}+51b^{19}}{a^{19}+b^{19}}\)
Approach Solution -2
The correct answer is the option (D) 51
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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