From the equestion whose roots are 1,3-√-2
Answers
Answer:
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 = - ba and αβ = ca.
Now, ax2 + bx + c = 0
⇒ x2 + bax + ca = 0 (Since, a ≠ 0)
⇒ x2 - (α + β)x + αβ = 0, [Since, α + β = -ba and αβ = ca]
⇒ x2 - (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
Solved examples to form the quadratic equation whose roots are given:
1. Form an equation whose roots are 2, and - 12.
Solution:
The given roots are 2 and -12.
Therefore, sum of the roots, S = 2 + (-12) = 32
And tghe product of the given roots, P = 2 ∙ -12 = - 1.
Therefore, the required equation is x2 – Sx + p
i.e., x2 - (sum of the roots)x + product of the roots = 0
i.e., x2 - 32x – 1 = 0
i.e, 2x2 - 3x - 2 = 0
2. Find the quadratic equation with rational coefficients which has 13+2√2 as a root.
Solution:
According to the problem, coefficients of the required quadratic equation are rational and its one root is 13+2√2 = 13+2√2 ∙ 3−2√23−2√2 = 3−2√29−8 = 3 - 2√2.
We know in a quadratic with rational coefficients irrational roots occur in conjugate pairs).
Since equation has rational coefficients, the other root is 3 + 2√2.
Now, the sum of the roots of the given equation S = (3 - 2√2) + (3 + 2√2) = 6
Product of the roots, P = (3 - 2√2)(3 + 2√2) = 32 - (2√2)2 = 9 - 8 = 1
Hence, the required equation is x2 - Sx + P = 0 i.e., x2 - 6x + 1 = 0.
2. Find the quadratic equation with real coefficients which has -2 + i as a root (i = √-1).
Solution:
According to the problem, coefficients of the required quadratic equation are real and its one root is -2 + i.
We know in a quadratic with real coefficients imaginary roots occur in conjugate pairs).
Since equation has rational coefficients, the other root is -2 - i
Now, the sum of the roots of the given equation S = (-2 + i) + (-2 - i) = -4
Product of the roots, P = (-2 + i)(-2 - i) = (-2)2 - i2 = 4 - (-1) = 4 + 1 = 5
Hence, the required equation is x2 - Sx + P = 0 i.e., x2 - 4x + 5 = 0.
Step-by-step explanation: