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If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q?
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If the zeroes of the polynomial x ^ 2 + px + q are double in value to ...
Step 1: Identify the Roots of the Second Polynomial
To find the roots of the polynomial \(2x^2 - 5x - 3\), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = -5\), and \(c = -3\).
Calculating the Discriminant
- \(b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49\)
Finding the Roots
\[
x = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}
\]
- Roots: \(x_1 = 3\), \(x_2 = -\frac{1}{2}\)
Step 2: Determine the Roots of the First Polynomial
Given that the roots of \(x^2 + px + q\) are double the roots of the second polynomial:
- New roots: \(2x_1 = 6\) and \(2x_2 = -1\)
Step 3: Formulate the Polynomial
The polynomial with roots \(6\) and \(-1\) can be expressed as:
\[
x^2 - (6 - 1)x + (6)(-1)
\]
This simplifies to:
\[
x^2 - 5x - 6
\]
Step 4: Identify p and q
From the polynomial \(x^2 + px + q\):
- \(p = -5\)
- \(q = -6\)
Final Values
- \(p = -5\)
- \(q = -6\)
These are the values of \(p\) and \(q\) for the polynomial \(x^2 + px + q\).
To find the roots of the polynomial \(2x^2 - 5x - 3\), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \(a = 2\), \(b = -5\), and \(c = -3\).
Calculating the Discriminant
- \(b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49\)
Finding the Roots
\[
x = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}
\]
- Roots: \(x_1 = 3\), \(x_2 = -\frac{1}{2}\)
Step 2: Determine the Roots of the First Polynomial
Given that the roots of \(x^2 + px + q\) are double the roots of the second polynomial:
- New roots: \(2x_1 = 6\) and \(2x_2 = -1\)
Step 3: Formulate the Polynomial
The polynomial with roots \(6\) and \(-1\) can be expressed as:
\[
x^2 - (6 - 1)x + (6)(-1)
\]
This simplifies to:
\[
x^2 - 5x - 6
\]
Step 4: Identify p and q
From the polynomial \(x^2 + px + q\):
- \(p = -5\)
- \(q = -6\)
Final Values
- \(p = -5\)
- \(q = -6\)
These are the values of \(p\) and \(q\) for the polynomial \(x^2 + px + q\).
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If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q?
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If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q?.
If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If the zeroes of the polynomial x ^ 2 + px + q are double in value to the zerves of 2x ^ 2 - 5x - 3 find the value of p and q?.
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