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If difference between the roots ofthe equation x2 – kx + 8 = 0 is 4 then the value of K is:
- a)0
- b)±4
- c)±8√3
- d)±4√3
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
If difference between the roots ofthe equation x2– kx + 8 = 0 is...
let α, β are roots of x2 – kx + 8 = 0
∴ α + β = -b/a = −(−k)/1 = k & α. β = c/a = 8/1 = 8
(α – β)2 = (α + β)2 – 4αβ = 42
⇒ k2 – 4 × 8 = 16
or k2 = 48 ⇒ k = ±√16×3 ⇒ k = ±4√3
(d) is correct.
∴ α + β = -b/a = −(−k)/1 = k & α. β = c/a = 8/1 = 8
(α – β)2 = (α + β)2 – 4αβ = 42
⇒ k2 – 4 × 8 = 16
or k2 = 48 ⇒ k = ±√16×3 ⇒ k = ±4√3
(d) is correct.
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If difference between the roots ofthe equation x2– kx + 8 = 0 is...
Understanding the Problem
We need to find the value of k in the quadratic equation x² - kx + 8 = 0, given that the difference between its roots is 4.
Quadratic Roots Formula
The roots of a quadratic equation ax² + bx + c = 0 can be found using the formula:
- Roots = [ -b ± √(b² - 4ac) ] / 2a
For our equation:
- a = 1
- b = -k
- c = 8
Difference Between Roots
The difference between the roots (let's call them r1 and r2) can be expressed as:
- r1 - r2 = √(b² - 4ac) / a
In this equation, we know that the difference is given as 4. Thus:
- (r1 - r2) = 4 = √(k² - 32)
Setting Up the Equation
To find k, we square both sides to eliminate the square root:
- 4² = k² - 32
- 16 = k² - 32
Now, rearranging gives us:
- k² = 16 + 32
- k² = 48
Finding k
Taking the square root of both sides gives:
- k = ±√48
- k = ±4√3
Conclusion
The possible values of k that satisfy the condition of the problem are:
- k = ±4√3
Thus, the correct answer is option 'D'.
We need to find the value of k in the quadratic equation x² - kx + 8 = 0, given that the difference between its roots is 4.
Quadratic Roots Formula
The roots of a quadratic equation ax² + bx + c = 0 can be found using the formula:
- Roots = [ -b ± √(b² - 4ac) ] / 2a
For our equation:
- a = 1
- b = -k
- c = 8
Difference Between Roots
The difference between the roots (let's call them r1 and r2) can be expressed as:
- r1 - r2 = √(b² - 4ac) / a
In this equation, we know that the difference is given as 4. Thus:
- (r1 - r2) = 4 = √(k² - 32)
Setting Up the Equation
To find k, we square both sides to eliminate the square root:
- 4² = k² - 32
- 16 = k² - 32
Now, rearranging gives us:
- k² = 16 + 32
- k² = 48
Finding k
Taking the square root of both sides gives:
- k = ±√48
- k = ±4√3
Conclusion
The possible values of k that satisfy the condition of the problem are:
- k = ±4√3
Thus, the correct answer is option 'D'.
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