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

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.

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)
= c⁄a

Product of the roots = αβ = c⁄a = (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

(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 = -b⁄a = -(-p)⁄1 = p Ans.
Product of the roots = c⁄a = 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 = -b⁄a = -(lm)⁄ l = -m Ans.
Product of the roots = c⁄a = lmn⁄l = mn Ans.
Great Deals on School & Homeschool Curriculum Books

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 = -b⁄a = -(-5)⁄1 = 5; ..........(i)
2(β) = c⁄a = 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.

Progressive Learning of Math : Quadratic Formula

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.

Answers to Exercise : Quadratic Formula

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