Advertisements
Advertisements
प्रश्न
If roots α, β of the equation \[x^2 - px + 16 = 0\] satisfy the relation α2 + β2 = 9, then write the value P.
उत्तर
Given equation: \[x^2 - px + 16 = 0\]
Also,
\[\alpha\] and \[\beta\] are the roots of the equation satisfying \[\alpha^2 + \beta^2 = 9 .\]
From the equation, we have:
Sum of the roots = \[\alpha + \beta =\]\[- \left( \frac{- p}{1} \right) = p\]
Product of the roots = \[\alpha\beta\] = \[\frac{16}{1} = 16\]
\[\text { Now }, \left( \alpha + \beta \right)^2 = \alpha^2 + \beta^2 + 2\alpha\beta\]
\[ \Rightarrow p^2 = 9 + 32\]
\[ \Rightarrow p^2 = 41\]
\[ \Rightarrow p = \sqrt{41}\]
Hence, the value of \[p \text { is } \sqrt{41} .\]
APPEARS IN
संबंधित प्रश्न
Solve the equation x2 + 3 = 0
Solve the equation 2x2 + x + 1 = 0
Solve the equation x2 + 3x + 9 = 0
Solve the equation –x2 + x – 2 = 0
Solve the equation x2 + 3x + 5 = 0
Solve the equation 27x2 – 10x + 1 = 0
Solve the equation 21x2 – 28x + 10 = 0
9x2 + 4 = 0
x2 + 2x + 5 = 0
\[x^2 + x + 1 = 0\]
\[17 x^2 - 8x + 1 = 0\]
\[17 x^2 + 28x + 12 = 0\]
\[21 x^2 - 28x + 10 = 0\]
\[8 x^2 - 9x + 3 = 0\]
\[\sqrt{2} x^2 + x + \sqrt{2} = 0\]
\[x^2 + x + \frac{1}{\sqrt{2}} = 0\]
\[x^2 + \frac{x}{\sqrt{2}} + 1 = 0\]
\[- x^2 + x - 2 = 0\]
\[3 x^2 - 4x + \frac{20}{3} = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 5 - i \right) x + \left( 18 + i \right) = 0\]
Solve the following quadratic equation:
\[i x^2 - x + 12i = 0\]
Write the number of quadratic equations, with real roots, which do not change by squaring their roots.
If α, β are roots of the equation \[x^2 - a(x + 1) - c = 0\] then write the value of (1 + α) (1 + β).
The complete set of values of k, for which the quadratic equation \[x^2 - kx + k + 2 = 0\] has equal roots, consists of
For the equation \[\left| x \right|^2 + \left| x \right| - 6 = 0\] ,the sum of the real roots is
If a, b are the roots of the equation \[x^2 + x + 1 = 0, \text { then } a^2 + b^2 =\]
If α, β are roots of the equation \[4 x^2 + 3x + 7 = 0, \text { then } 1/\alpha + 1/\beta\] is equal to
If α, β are the roots of the equation \[a x^2 + bx + c = 0, \text { then } \frac{1}{a\alpha + b} + \frac{1}{a\beta + b} =\]
The number of solutions of `x^2 + |x - 1| = 1` is ______.
If x is real and \[k = \frac{x^2 - x + 1}{x^2 + x + 1}\], then
The value of a such that \[x^2 - 11x + a = 0 \text { and } x^2 - 14x + 2a = 0\] may have a common root is
The values of k for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are
If the equations \[x^2 + 2x + 3\lambda = 0 \text { and } 2 x^2 + 3x + 5\lambda = 0\] have a non-zero common roots, then λ =
If α, β are the roots of the equation \[x^2 + px + q = 0 \text { then } - \frac{1}{\alpha} + \frac{1}{\beta}\] are the roots of the equation
If α, β are the roots of the equation \[x^2 - p(x + 1) - c = 0, \text { then } (\alpha + 1)(\beta + 1) =\]
The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
If 1 – i, is a root of the equation x2 + ax + b = 0, where a, b ∈ R, then find the values of a and b.
