QUADRATIC FORMULA - RELATION BETWEEN ROOTS AND COEFFICIENTS, EXAMPLES, EXERCISE

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Please study
Derivation of Quadratic Formula
if you have not already done so.

There, We presented the Derivation
Of The Formula in a lucid way.
An Example in applying the formula
is given. Another problem for practice
is also given.

Here we derive the Relation between roots
and coefficients of a Quadratic Equation.

we present some Solved Examples on these relations.

We also give problems for practice in Exercise.



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Relation between roots and coefficients of a Quadratic Equation

Let the roots of ax2 + bx + c = 0 be
α (called alpha) and β (called beta).
Then By Quadratic Formula
α = {-b + √(b2 - 4ac)}⁄2a and β = {-b - √(b2 - 4ac)}⁄2a

Sum of the roots = α + β
= {-b + √(b2 - 4ac)}⁄2a + {-b - √(b2 - 4ac)}⁄2a
= {-b + √(b2 - 4ac) -b - √(b2 - 4ac)}⁄2a
= {-2b}⁄2a = -b⁄a

Sum of the roots = α + β = -b⁄a = -{(coefficient of x)⁄(coefficient of x2)}

Product of the roots = α x β
= {-b + √(b2 - 4ac)}⁄2a x {-b - √(b2 - 4ac)}⁄2a
= [{-b + √(b2 - 4ac)} x {-b - √(b2 - 4ac)}]⁄(4a2)
The Numerator is product of sum and difference of two terms
which we knowis equal to the difference of the squares of the two terms.

∴ Product of the roots = αβ = [(-b)2 - {√(b2 - 4ac)}2]⁄(4a2)
= [b2 - (b2 - 4ac)]⁄(4a2) = [b2 - b2 + 4ac)]⁄(4a2) = (4ac)⁄(4a2)
= ca

Product of the roots = αβ = ca = (constant term)⁄(coefficient of x2)}

Example 1 : Quadratic Formula

Find the sum and product of the roots of the equations given below.

  1. 3x2 + 2x + 1 = 0
  2. x2 - px + pq = 0
  3. lx2 + lmx + lmn = 0


Solution of Example 1 of Quadratic Formula :

(i) The given equation is 3x2 + 2x + 1 = 0
Comparing this equation with ax2 + bx + c = 0, we get
a = 3, b = 2 and c = 1
We know, Sum of the roots = -ba = -2⁄3 Ans.
Product of the roots = ca = 1⁄3

(ii) The given equation is x2 - px + pq = 0
Comparing this equation with ax2 + bx + c = 0, we get
a = 1, b = -p and c = pq
We know, Sum of the roots = -ba = -(-p)⁄1 = p Ans.
Product of the roots = ca = pq ⁄1 = pq Ans.

(iii) The given equation is lx2 + lmx + lmn = 0
Comparing this equation with ax2 + bx + c = 0, we get
a = l, b = lm and c = lmn
We know, Sum of the roots = -ba = -(lm)⁄ l = -m Ans.
Product of the roots = ca = lmnl = mn Ans.
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Example 2 : Quadratic Formula

If one root of x2 - 5x + k = 0 is 2, find the value of kand the other root.

Solution of Example 2 of Quadratic Formula :

The given equation is x2 - 5x + k = 0
Comparing this equation with ax2 + bx + c = 0, we get
a = 1, b = -5 and c = k
By data one root is 2. Let the other root be β
Then 2 + β = -ba = -(-5)⁄1 = 5; ..........(i)
2(β) = ca = k⁄1 = k ⇒ β = k⁄2 ......(ii)
Using (ii) in (i), we get 2 + k⁄2 = 5 ⇒ k⁄2 = 5 - 2 = 3
k = 3 x 2 = 6. Ans.
The other root = β = k⁄2 = 6⁄2 = 3. Ans.

Exercise : Quadratic Formula

  1. Find the sum and product of the roots of the equations given below.
    1. px2 - rx + q = 0
    2. x2 - px + q = 0
    3. 9x2 + 4x - 11 = 0
  2. If one root of x2 -(p - 1)x + 10 = 0 is 5, then findthe value of p and the second root.

For Answers See at the bottom of the Page.

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Answers to Exercise : Quadratic Formula

    1. rq, qp
    2. p, q
    3. -4⁄9, -11⁄9
  1. p = 8, second root = 2