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Mathematics

Relation Between the Roots and Coefficients of a Quadratic Equations

Gk Scientist May 6, 2022 No Comments
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Roots and Coefficients of a Quadratic Equations:

The roots of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) are given by,

Relation Between the Roots and Coefficients of a Quadratic Equations - Relation Between the Roots and Coefficients of a Quadratic Equations
Example 1- If one of the roots of the equation ax2 + bx + c = 0 is three times the other, then prove that 3b2 = 16 ac.

Solution- The given equation is,
ax2 + bx + c = 0

Let α, 3α be its roots.

Now, α + 3α = -b/a
⇒ 4α = -b/a
⇒ α = -b/4a …………(i)

Also, α . 3α = c/a
⇒ 3α2 = c/a
⇒ α2 = c/3a …………(ii)

Substitute the value of α from (i) in (ii)-
(-b/4a)2 = c/3a
⇒ b2/16a2 = c/3a
⇒ 3b2 = 16ac
Example 2- If αβ are the roots of the equation x2 – 6x + P = 0 such that 3α + 2β = 20, then find the value of P.

Solution- The given equation is,
x2 – 6x + P = 0

Now, α, and β are their roots.
α + β = 6 ……….(i)
and αβ = P ……….(ii)
Also 3α + 2β = 20 ……….(iii)

Solving (i) and (iii)-
α + β = 6
3α + 2β = 20

⇒ α/(20 – 12) = β/(18 – 20) = -1/(2 – 3)
⇒ α/8 = β/(-2) = 1/1
⇒ α = 8, β = -2

Now from (ii)-
αβ = P
⇒ (8) (-2) = P
⇒ P = -16
Example 3- If the difference of the roots of equation x2 + ax + b = 0 be unity then prove that a2 + 4b2 = (1 + 2b)2.

Solution- The given equation is,
x2 + ax + b = 0

Let its roots be α and β such that α – β = 1
⇒ √[(α + β)2 – (4αβ)] = 1
⇒ √[(-a)2 – 4b] = 1
Squaring both sides, we get-
⇒ a2 – 4b = 1
⇒ a2 = 1 + 4b
⇒ a2 + 4b2 = 1 + 4b + 4b2
⇒ a2 + 4b2 = (1 + 2b)2
Example 4- If k be the ratio of the roots of equation x2 – ax + b = 0 then prove that (k2 + 1)/k = (a2 – 2b)/b.

Solution- The given equation is,
x2 – ax + b = 0

Let αk and α be the roots of the equation.

Sum of root = -b/a
⇒ αk + α = a
⇒ α (k + 1) = a
⇒ α = a/(k + 1) ……….(i)

Product of root = c/a
⇒ αk . α = b
⇒ α2k = b ……….(ii)

Substitute (i) in (ii), we get-
[a/(k + 1)]2 k = b
⇒ [a2/(k2 + 1 + 2k)] k = b
⇒ a2k = b (k2 + 1 + 2k)
⇒ b (k2 + 1) = a2k – 2bk
⇒ b (k2 + 1) = k (a2 – 2b)
⇒ (k2 + 1)/k = (a2 – 2b)/b

Complex Numbers Operations
Square Roots of Complex Numbers
Cube Roots of Unity
Demoivre’s Theorem
Representation of Complex Numbers
Representation of a Complex Number in Polar Form
Geometrical Representation of Fundamental Operations of Complex Numbers
Complex number– Wikipedia
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