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 ax^{2} + bx + c = 0 be α (called alpha) and β (called beta). Then By Quadratic Formula α = {-b + √(b^{2} - 4ac)}⁄2a and β = {-b - √(b^{2} - 4ac)}⁄2a

Sum of the roots = α + β = -b⁄a = -{(coefficient of x)⁄(coefficient of x^{2})}

Product of the roots = α x β = {-b + √(b^{2} - 4ac)}⁄2a x {-b - √(b^{2} - 4ac)}⁄2a = [{-b + √(b^{2} - 4ac)} x {-b - √(b^{2} - 4ac)}]⁄(4a^{2}) 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 = c⁄a = (constant term)⁄(coefficient of x^{2})}

Example 1 : Quadratic Formula

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

3x^{2} + 2x + 1 = 0

x^{2} - px + pq = 0

lx^{2} + lmx + lmn = 0

Solution of Example 1 of Quadratic Formula :

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

(ii) The given equation is x^{2} - px + pq = 0 Comparing this equation with ax^{2} + 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 lx^{2} + lmx + lmn = 0 Comparing this equation with ax^{2} + 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 x^{2} - 5x + k = 0 is 2, find the value of kand the other root.

Solution of Example 2 of Quadratic Formula :

The given equation is x^{2} - 5x + k = 0 Comparing this equation with ax^{2} + 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

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

px^{2} - rx + q = 0

x^{2} - px + q = 0

9x^{2} + 4x - 11 = 0

If one root of x^{2} -(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

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