find a quadratic polynomial, whose zeroes are √2 and -3/2
Answers
EXPLANATION.
Quadratic polynomial whose zeroes are : √2 and -3√2.
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ √2 + (-3√2) = √2 - 3√2. = -2√2.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ (√2)(-3√2) = -6.
As we know that,
Equation of quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Put the values in the equation, we get.
⇒ x² - (-2√2)x + (-6) = 0.
⇒ x² + 2√2x - 6 = 0.
MORE INFORMATION.
Conjugate roots.
(1) = If D < 0.
One roots = α + iβ.
Other roots = α - iβ.
(2) = If D > 0.
One roots = α + √β.
Other roots = α - √β.
Answer :-
- Required Quadratic polynomial : x²+2√2x-6=0
Step by step explanation :-
Given :-
- 2 zeroes of the quadratic polynomial = √2 and -3√2.
To find :-
- Quadratic Polynomial.
Concept :-
• Firstly, adding up the given zeroes and find a particular value.
• Then, multiplying the zeroes and finding a value.
• Then, by applying the formula for finding the quadratic polynomial when the sum and product of zeroes is given, the required answer can obtained.
Solution :-
Given, 2 zeroes = √2 and -3√2.
let the first zero ( √2 ) be α
let the second zero ( -3√2 ) be β
Now, Sum of the zeroes :-
⠀⠀⠀⠀⠀⠀⠀⠀⠀α+β
⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ -3√2+√2
⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ -2√2
And, Product of the zeroes :-
⠀⠀⠀⠀⠀⠀⠀⠀⠀αβ
⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ -3√2 × √2
⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ -3 × 2
⠀⠀⠀⠀⠀⠀⠀⠀⠀⇒ -6
We know that,
Formula for finding the quadratic polynomial :-
⠀⠀⠀p(x) = x²-(sum of zeroes)x + (product of zeroes)
⠀⠀⠀⠀⠀⇒ x²-(-2√2)x + (-6)
⠀⠀⠀⠀⠀⇒ x²+2√2x-6 ✓