if one zero of polynomial (k^2+16)x^2+16x+8k is reciprocal of the other then k is equal to
Answers
EXPLANATION.
One zeroes of polynomial,
⇒ (k² + 16)x² + 16x + 8k is reciprocal of the other.
As we know that,
Let one zeroes be = α.
Other zeroes be = 1/α.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ α x 1/α = 8k/(k² + 16).
⇒ 1 = 8k/k² + 16.
⇒ k² + 16 = 8k.
⇒ k² - 8k + 16 = 0.
As we know that,
Factorizes the equation into middle term splits, we get.
⇒ k² - 4k - 4k + 16 = 0.
⇒ k(k - 4) - 4(k - 4) = 0.
⇒ (k - 4)(k - 4) = 0.
⇒ (k - 4)² = 0.
⇒ (k - 4) = 0.
⇒ k = 4.
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.
~ We have to find out the value of k if one zero of polynomial (k²+16)x²+16x+8k is reciprocal of the other.
~ Assumptions:
~ Using concept:
As we know that αβ = c/a is used to find out the product of zeroes of the polynomial.
~ Solution firstly by using the above formula let us find out the
»»» a × 1/a = 8k/(k²+16)
»»» a/1 × 1/a = 8k/(k²+16)
»»» 1 = 8k/(k²+16) [a cancel's a]
»»» 1 = 8k/k²+16
»»» k²+16 = 8k [- = + ; + = -]
»»» k² - 8k + 16 = 0
~ Now let's factorise the expression by using middle term splitting method.
»»» k² - 8k + 16 = 0
»»» k²-4k-4k+16 = 0
»»» k(k) - (4) - 4(k)(-4) = 0
»»» k(k-4) - 4(k-4) = 0
»»» (k-4) (k-4) = 0
»»» k = 0+4 or k = 0+4
»»» k = 4 or k = 4
»»» k = 4
Henceforth, 4 is the value of k.
Some knowledge about Quadratic Equations -
★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a
★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a
★ Discriminant is given by b²-4ac
- Discriminant tell us about there are solution of a quadratic equation as no solution, one solution and two solutions.
★ A quadratic equation have 2 roots
★ ax² + bx + c = 0 is the general form of quadratic equation
★ D > 0 then roots are real and distinct.
★ D = 0 then roots are real and equal.
★ D < 0 then roots are imaginary.
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