The parallel and geometric mean of the roots of a quadratic equation is 8 and 5. Find the quadratic equation.
The parallel and geometric mean of the roots of a quadratic equation is 8 and 5. Find the quadratic equation.
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Step-by-step explanation:
Let the two roots be α and β
∴
2
α+β
=8 (AM of a and b is
2
a+b
) ⟶(1)
and
αβ
=5 (GM of a and b is
ab
) ⟶(2)
From (1)
α+β=16
⇒ sum of roots =α+β=16
From (2)
αβ
=5
⇒αβ=25
∴ Products of roots =25
Now, we know that if we have the sum of roots and product of roots then the quadratic equation is given by
x
2
− (sum of the roots)x + (product of the roots) =0
∴ the quadratic equation with roots α,β are
x
2
−(α+β)x+αβ=0
x
2
−16x+25=0