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If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.?
Most Upvoted Answer
If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quad...
2 is a root if the equation 3x^2 +px-8=0.`. 3(2)^2+p(2)-8=0 =》12+2p-8=0=》4+2p=0=》p=-2Now,4x^2-2px+k=0 =》4x^2+4x+k=0 It has equal roots .`.D=0 =》(4)^2-4×4×k=0=》16-16k=0=》16(1-k)=0=》1-k=0=》k=1 Hence, k= 1
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If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quad...
Solution:
We are given that 2 is a root of the quadratic equation 3x² px - 8 = 0.
Let us use the factor theorem to factorize this quadratic equation.
Factor Theorem: If c is a root of the polynomial f(x), then (x-c) is a factor of f(x).
Since 2 is a root of 3x² px - 8 = 0, we can say that (x-2) is a factor of 3x² px - 8 = 0.
We can use this information to find the value of p.
Using Factor Theorem:
Let f(x) = 3x² px - 8
Since (x-2) is a factor of f(x), we can say that f(2) = 0.
f(2) = 3(2)² p(2) - 8
0 = 12 - 2p - 8
0 = 4 - 2p
2 = p
We have found the value of p.
Now, let us use this value of p to find the value of k in the quadratic equation 4x² - 2px k = 0.
Using Discriminant:
The quadratic equation 4x² - 2px k = 0 has equal roots.
This means that the discriminant of the quadratic equation is zero.
The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.
If the discriminant is zero, then b² - 4ac = 0.
In our case, the quadratic equation is 4x² - 2px k = 0.
So, a = 4, b = -2p and c = k.
Substituting these values in the discriminant, we get:
(-2p)² - 4(4)(k) = 0
4p² - 16k = 0
4p² = 16k
k = (4p²)/16
k = p²/4
k = 1
Therefore, the value of k is 1.
We are given that 2 is a root of the quadratic equation 3x² px - 8 = 0.
Let us use the factor theorem to factorize this quadratic equation.
Factor Theorem: If c is a root of the polynomial f(x), then (x-c) is a factor of f(x).
Since 2 is a root of 3x² px - 8 = 0, we can say that (x-2) is a factor of 3x² px - 8 = 0.
We can use this information to find the value of p.
Using Factor Theorem:
Let f(x) = 3x² px - 8
Since (x-2) is a factor of f(x), we can say that f(2) = 0.
f(2) = 3(2)² p(2) - 8
0 = 12 - 2p - 8
0 = 4 - 2p
2 = p
We have found the value of p.
Now, let us use this value of p to find the value of k in the quadratic equation 4x² - 2px k = 0.
Using Discriminant:
The quadratic equation 4x² - 2px k = 0 has equal roots.
This means that the discriminant of the quadratic equation is zero.
The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac.
If the discriminant is zero, then b² - 4ac = 0.
In our case, the quadratic equation is 4x² - 2px k = 0.
So, a = 4, b = -2p and c = k.
Substituting these values in the discriminant, we get:
(-2p)² - 4(4)(k) = 0
4p² - 16k = 0
4p² = 16k
k = (4p²)/16
k = p²/4
k = 1
Therefore, the value of k is 1.
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If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.?.
If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If 2 is a root of the quadratic equation 3x² + px - 8 = 0 and the quadratic equation 4x² - 2px + k = 0 has equal roots, find the value of k.?.
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