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If P and Q are the roots of f(x) = x2 - 14x + 45, then find the value of (1/P +1/Q)
- a)45/14
- b)14/45
- c)41/54
- d)54/41
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If P and Q are the roots of f(x) = x2- 14x + 45, then find the value o...
To find the value of (1/P + 1/Q), we need to determine the values of P and Q first.
Given that P and Q are the roots of the quadratic equation f(x) = x^2 - 14x + 45, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -14, and c = 45. Plugging these values into the quadratic formula, we get:
P = (-(-14) ± √((-14)^2 - 4(1)(45))) / (2(1))
= (14 ± √(196 - 180)) / 2
= (14 ± √16) / 2
= (14 ± 4) / 2
= (18 / 2) or (10 / 2)
= 9 or 5
So, P can have the value of either 9 or 5.
Now, let's find the value of Q. Since P and Q are the roots of the quadratic equation, if P = 9, then Q = 5, and vice versa.
Now, we can calculate (1/P + 1/Q) using the values of P and Q that we found.
(1/P + 1/Q) = (1/9 + 1/5)
To add these fractions, we need a common denominator. The least common denominator of 9 and 5 is 45. We can rewrite the fractions with the common denominator:
(1/P + 1/Q) = (5/45 + 9/45)
Now, we can add the fractions:
(1/P + 1/Q) = (5 + 9) / 45
= 14 / 45
Therefore, the value of (1/P + 1/Q) is 14/45, which corresponds to option B.
Given that P and Q are the roots of the quadratic equation f(x) = x^2 - 14x + 45, we can use the quadratic formula to find their values.
The quadratic formula states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -14, and c = 45. Plugging these values into the quadratic formula, we get:
P = (-(-14) ± √((-14)^2 - 4(1)(45))) / (2(1))
= (14 ± √(196 - 180)) / 2
= (14 ± √16) / 2
= (14 ± 4) / 2
= (18 / 2) or (10 / 2)
= 9 or 5
So, P can have the value of either 9 or 5.
Now, let's find the value of Q. Since P and Q are the roots of the quadratic equation, if P = 9, then Q = 5, and vice versa.
Now, we can calculate (1/P + 1/Q) using the values of P and Q that we found.
(1/P + 1/Q) = (1/9 + 1/5)
To add these fractions, we need a common denominator. The least common denominator of 9 and 5 is 45. We can rewrite the fractions with the common denominator:
(1/P + 1/Q) = (5/45 + 9/45)
Now, we can add the fractions:
(1/P + 1/Q) = (5 + 9) / 45
= 14 / 45
Therefore, the value of (1/P + 1/Q) is 14/45, which corresponds to option B.
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If P and Q are the roots of f(x) = x2- 14x + 45, then find the value o...
GIVEN:
f(x) = x2 - 14x + 45
FORMULA USED:
Sum of roots = (-b/a) and Product of roots = c/a for f(x) = ax2 + bx + c
CALCULATION:
f(x) = x2 - 14x + 45
⇒ a = 1, b = -14, c = 45
⇒ Sum of roots(P + Q) = (-b/a)
⇒ 14
⇒ Product of roots (PQ) = c/a
⇒ 45
⇒ (1/P + 1/Q) = (P + Q)/PQ
⇒ 14/45
∴ (1/P + 1/Q) = 14/45
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If P and Q are the roots of f(x) = x2- 14x + 45, then find the value of (1/P +1/Q)a)45/14b)14/45c)41/54d)54/41Correct answer is option 'B'. Can you explain this answer?
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