Advertisements
Advertisements
प्रश्न
If the difference of the roots of \[x^2 - px + q = 0\] is unity, then
विकल्प
\[p^2 + 4q = 1\]
\[p^2 - 4q = 1\]
\[p^2 + 4 q^2 = (1 + 2q )^2\]
\[4 p^2 + q^2 = (1 + 2p )^2\]
उत्तर
\[p^2 - 4q = 1\]
Given equation:
\[x^2 - px + q = 0\]
Also
\[\alpha \text { and } \beta\] are the roots of the equation such that \[\alpha - \beta = 1\].
Sum of the roots = \[\alpha + \beta = \frac{- \text { Coefficient of } x}{\text { Coefficient of } x^2} = - \left( \frac{- p}{1} \right) = p\]
Product of the roots = \[\alpha\beta = \frac{\text { Constant term }}{\text { Coefficient of } x^2} = q\]
\[\therefore (\alpha + \beta )^2 - (\alpha - \beta )^2 = 4\alpha\beta\]
\[ \Rightarrow p^2 - 1 = 4q\]
\[ \Rightarrow p^2 - 4q = 1\]
APPEARS IN
संबंधित प्रश्न
Solve the equation 2x2 + x + 1 = 0
Solve the equation x2 + 3x + 9 = 0
Solve the equation –x2 + x – 2 = 0
Solve the equation `sqrt2x^2 + x + sqrt2 = 0`
Solve the equation `x^2 + x + 1/sqrt2 = 0`
Solve the equation `x^2 + x/sqrt2 + 1 = 0`
Solve the equation `x^2 -2x + 3/2 = 0`
x2 + x + 1 = 0
\[21 x^2 + 9x + 1 = 0\]
\[17 x^2 - 8x + 1 = 0\]
\[17 x^2 + 28x + 12 = 0\]
\[3 x^2 - 4x + \frac{20}{3} = 0\]
Solving the following quadratic equation by factorization method:
\[6 x^2 - 17ix - 12 = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 3\sqrt{2} + 2i \right) x + 6\sqrt{2i} = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 2 + i \right) x - \left( 1 - 7i \right) = 0\]
Solve the following quadratic equation:
\[2 x^2 + \sqrt{15}ix - i = 0\]
Solve the following quadratic equation:
\[i x^2 - x + 12i = 0\]
If the difference between the roots of the equation \[x^2 + ax + 8 = 0\] is 2, write the values of a.
Write roots of the equation \[(a - b) x^2 + (b - c)x + (c - a) = 0\] .
If α, β are roots of the equation \[x^2 + lx + m = 0\] , write an equation whose roots are \[- \frac{1}{\alpha}\text { and } - \frac{1}{\beta}\].
If α, β are roots of the equation \[x^2 - a(x + 1) - c = 0\] then write the value of (1 + α) (1 + β).
If a, b are the roots of the equation \[x^2 + x + 1 = 0, \text { then } a^2 + b^2 =\]
The number of real roots of the equation \[( x^2 + 2x )^2 - (x + 1 )^2 - 55 = 0\] is
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 the roots of \[x^2 - bx + c = 0\] are two consecutive integers, then b2 − 4 c 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
The value of p and q (p ≠ 0, q ≠ 0) for which p, q are the roots of the equation \[x^2 + px + q = 0\] are
The set of all values of m for which both the roots of the equation \[x^2 - (m + 1)x + m + 4 = 0\] are real and negative, is
If α and β are the roots of \[4 x^2 + 3x + 7 = 0\], then the value of \[\frac{1}{\alpha} + \frac{1}{\beta}\] is
If α, β are the roots of the equation \[x^2 - p(x + 1) - c = 0, \text { then } (\alpha + 1)(\beta + 1) =\]
The least value of k which makes the roots of the equation \[x^2 + 5x + k = 0\] imaginary is
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.
If `|(z - 2)/(z + 2)| = pi/6`, then the locus of z is ______.
