If alpha and beta are the roots of the equation x^2+7x+12=0 then equation whose roots are (alpha+beta) and (alpha*beta)
Answers
If alpha and beta are the roots of the equation x^2 + 7x + 12 = 0, then equation whose roots are (alpha+beta) and (alpha*beta) is
x^2 - 5x - 84 = 0
Given:
alpha and beta are the roots of the equation x^2+7x+12=0
To find:
Equation whose roots are (alpha + beta) and (alpha * beta)
Solution:
For a general Quadratic equation: ax^2 + bx + c = 0; r1 and r2 being its two roots, then the relation between its roots is given as
Sum of roots = r1 + r2 = -b/a
Product of roots = r1 * r2 = c/a
Hence,
for the equation x^2 + 7x + 12=0, since alpha and beta are its two roots, we can say
alpha + beta = -7/1 = -7 ---(1)
alpha * beta = 12/1 = 12 ---(2)
Applying the relation that a quadratic equation, whose roots are u and v, can be written as x^2 - (u + v) x + u*v = 0,
equation whose roots are (alpha + beta) and (alpha * beta) can be written as
x^2 - ((alpha + beta) + (alpha * beta)) x + ((alpha + beta) * (alpha * beta)) = 0
Substituting the values of alpha + beta and alpha * beta from equations (1) and (2) in this expression, we get
x^2 - ((-7) + (12)) x + ((-7) * (12)) = 0
=> x^2 - (-7 + 12) x + ((-7) * (12)) = 0
=> x^2 - (5) x + (-84) = 0
=> x^2 - 5x - 84 = 0
Hence,
equation whose roots are (alpha + beta) and (alpha * beta) is given as: x^2 - 5x - 84 = 0
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