If α and β are the roots of equation x 2– x + 1 = 0, then which equation will have roots α 3and β 3?
x2 + 2x + 1 = 0
For \(x^2-x+1=0\), by Vieta's: \(\alpha+\beta=1\), \(\alpha\beta=1\).
Step 1 — Sum of cubes:
\[\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)=1-3=-2\]
Step 2 — Product of cubes: \(\alpha^3\beta^3=(\alpha\beta)^3=1\).
Step 3 — Form the equation: \(x^2-(\text{sum})x+(\text{product})=0\):
\[x^2-(-2)x+1=0 \implies x^2+2x+1=0\]
Therefore the required equation is x² + 2x + 1 = 0.
Similar Questions
If 2x 2+ 5x + 1 = 0, then one of the values of \(x - \frac{1}{{2x}}\) is:
Which of the following statement is correct?
I. If x = 12, y = -2 and z = -10, then x3 + y3 + z3 = 360.
II. If x + y = 48 and 4xy = 128, then 4x2 + 4y2 = 4480.
If \(4x^4-37x^2+9=0, x>\sqrt{\frac{3}{2}}\) , then what is the value of \(8x^3-\frac{27}{x^3}\)
If x 2 – 3x + 1 = 0, then the value of \(\frac{(x^4+\frac{1}{x^2})}{(x^2+5x+1)}\) is:
If x(2x + 3) = 90 and 7y -1/2 + 2y -1/2 = y 1/2 (x and y are positive numbers), then what is the value of x 2+ y 2?
A root of equation ax 2+ bx + c = 0 (where a, b and c are rational numbers) is 5 + 3√3. What is the value of (a 2+ b 2+ c 2)/(a + b + c)?
If [a + (1/a)] 2– 2[a – (1/a)] = 12, then which of the following is a value of 'a'?
If (x – 2) and (x + 3) are the factors of the equation x 2+ k 1x + k 2= 0, then what are the values of k 1 and k 2?
Sum of twice a fraction and its reciprocal is 17/6. What is the fraction?
Sum of a fraction and thrice of its reciprocal is 73/20. What is the fraction?
Important Questions from Quadratic Equation
For what values of k, the roots of 9x 2 + 8kx + 16 = 0 are real and equal?
- If \(\rm \left( \frac{x}{x+1} \right)^2 -5 \left( \frac{x}{x+1} \right) +6=0 \) , then the value of \(\rm \left( 1+\frac{1}{x} \right) \) is equal to :
The nature of the roots of the equation 4x 2 - 2x - 3 = 0.
If the roots of the equation (q – r)x 2+ (r – p)x + (p – q) = 0 are equal, then which of the following is true?
Number of real roots of the quadratic equation 3x 2+ 4x + 25 = 0 is
