Grade 11Algebra
If 'r' be the ratio of the roots of the equation ax^2 + bx + c = 0 , show that (r + 1)^2/r = b^2/ac
If 'r' be the ratio of the roots of the equation ax^2 + bx + c = 0 , show that
(r + 1)^2/r = b^2/ac
1 Answer
Arun
i) Let the two roots of the given quadratic equation be α & β;
then as given α/β = r; ==> α = β*r ---------- (1)
ii) By properties of roots of quadratic equation,
Sum of roots: α + β = -b/a;
Substituting from (1), α = β*r, β(1 + r) = -b/a;
Squaring, β²(1 + r)² = b²/a² --------- (2)
and Product of roots, α*β = c/a
Again substituting from (1) for α = β*r, β²*r = c/a ----- (3)
iii) Dividing (2) by (3): (1 + r)²/r = b²/ac
Cross multiplying, {(1 + r)²}*(ac) = (b²)*(r)
then as given α/β = r; ==> α = β*r ---------- (1)
ii) By properties of roots of quadratic equation,
Sum of roots: α + β = -b/a;
Substituting from (1), α = β*r, β(1 + r) = -b/a;
Squaring, β²(1 + r)² = b²/a² --------- (2)
and Product of roots, α*β = c/a
Again substituting from (1) for α = β*r, β²*r = c/a ----- (3)
iii) Dividing (2) by (3): (1 + r)²/r = b²/ac
Cross multiplying, {(1 + r)²}*(ac) = (b²)*(r)
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