How to form Quadratic Equation from Another Quadratic Equation 2

Formation of the Quadratic Equation whose Roots are Given

We will learn the formation of the quadratic equation whose roots are given.

To form a quadratic equation, let α and β be the two roots.

Let us assume that the required equation be ax² + bx + c = 0 (a ≠ 0).

According to the problem, roots of this equation are α and β.

Therefore,

α + β = - b/a and αβ = caca.

Now, ax² + bx + c = 0

⇒ x² + bx/a + c/a = 0 (Since, a ≠ 0)

⇒ x² - (α + β)x + αβ = 0,
[Since, α + β = -b/a and αβ = c/a]

⇒ x2² - (sum of the roots)x + product of the roots = 0

⇒ x² - Sx + P = 0, where S = sum of the roots and P = product of the roots .........(i)

Formula (i) is used for the formation of a quadratic equation when its roots are given.

For example suppose we are to form the quadratic equation whose roots are 5 and (-2). By formula (i) we get the required equation as

x² - [5 + (-2)]x + 5 ∙ (-2) = 0

⇒ x² - [3]x + (-10) = 0

⇒ x² - 3x - 10 = 0

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