QUADRATIC EQUATION SOLVER - FINDING THE QUADRATIC EQUATION WITH GIVEN ROOTS, EXAMPLES, EXERCISE

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Please study
Quadratic Formula before Quadratic Equation Solver
if you have not already done so.

There, We presented the Derivation Of
Formulae Relating roots and coefficients
of a Quadratic Equation.
Solved Examples on applying the formulae
are given. Problems for practice
are also given in Exercise.

That knowledge is a prerequisite here.

Here we deal with finding the Quadratic
Equation whose roots are given.

We explain the Formula and apply it
in solving problems. Exercise problems
are also given for practice.

To find the Quadratic Equation whose roots are given :

The Quadratic Equation whose roots are given is
x2 - (sum of the roots)x + (product of the roots) = 0

PROOF :

We know

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

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

Substituting these in the LHS of the Equation above
and multiplying through out with a
, we get the equation as
ax2 + bx + c = 0
which is the original Equation. (Proved.)

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Example 1 : Quadratic Equation Solver

Find the quadratic equation whose roots are 3, -2.

Solution of Example 1 of Quadratic Equation Solver :

The given roots are 3, -2
Sum of the roots = 3 + (-2) = 3 - 2 = 1; Product of the roots = 3 x (-2) = -6
We know the Quadratic Equation whose roots are given is
x2 - (sum of the roots)x + (product of the roots) = 0
∴ The required equation is x2 - (1)x + (-6) = 0
i.e x2 - x - 6 = 0 Ans.

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Example 2 : Quadratic Equation Solver

Find the quadratic equation whose roots are 4⁄3, 1⁄3.

Solution of Example 2 of Quadratic Equation Solver :

The given roots are 4⁄3, 1⁄3
Sum of the roots = 4⁄3 + 1⁄3 = 5⁄3; Product of the roots = 4⁄3 x 1⁄3 = 4⁄9
We know the Quadratic Equation whose roots are given is
x2 - (sum of the roots)x + (product of the roots) = 0
∴ The required equation is x2 - (5⁄3)x + (4⁄9) = 0
Multiplying both sides with 9, we get
9x2 -15x + 4 = 0 Ans.

Example 3 : Quadratic Equation Solver

Find the quadratic equation whose roots are lm, mn.

Solution of Example 3 of Quadratic Equation Solver :

The given roots are lm, mn
Sum of the roots = lm + mn = m(l + n); Product of the roots = lm x mn = lm2n
We know the Quadratic Equation whose roots are given is
x2 - (sum of the roots)x + (product of the roots) = 0
∴ The required equation is x2 - m(l + n)x + lm2n = 0. Ans.

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Example 4 : Quadratic Equation Solver

Find the quadratic equation whose roots are 5 + √7), (5 - √7).

Solution of Example 4 of Quadratic Equation Solver :

The given roots are 5 + √7, 5 - √7
Sum of the roots = (5 + √7) + (5 - √7) = 10;
Product of the roots = (5 + √7)x(5 - √7) = 52 - (√7)2 = 25 - 7 = 18
We know the Quadratic Equation whose roots are given is
x2 - (sum of the roots)x + (product of the roots) = 0
∴ The required equation is x2 - (10)x + (18) = 0
i.e x2 - 10x + 18 = 0 Ans.

Exercise : Quadratic Equation Solver

  1. Find the quadratic equation whose roots are -2, -4.
  2. Find the quadratic equation whose roots are 3 + √2, 3- √2
  3. Find the quadratic equation whose roots are (l - m), (l + m)
  4. Find the quadratic equation whose roots are 1/2, 3/2


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Answers to Exercise

  1. x2 + 6x + 8 = 0
  2. x2 - 6x + 7 = 0
  3. x2 - 2lx + (l2 - m2) = 0
  4. 4x2 - 8x + 3 = 0