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Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.?
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Find zeroes of a polynomial 6x²-18x+12 and verify the relationship bet...
To find the zeroes of the polynomial \(6x^2 - 18x + 12\) and verify the relationship between zeroes and coefficients, follow these steps:
Step 1: Finding the Zeroes
To find the zeroes, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \(a = 6\), \(b = -18\), and \(c = 12\).
- Calculate the discriminant (\(D\)):
\[
D = b^2 - 4ac = (-18)^2 - 4 \cdot 6 \cdot 12 = 324 - 288 = 36
\]
- Now, apply the quadratic formula:
\[
x = \frac{18 \pm \sqrt{36}}{2 \cdot 6} = \frac{18 \pm 6}{12}
\]
- This gives:
- \(x_1 = \frac{24}{12} = 2\)
- \(x_2 = \frac{12}{12} = 1\)
Thus, the zeroes are \(x = 2\) and \(x = 1\).
Step 2: Relationship Between Zeroes and Coefficients
For a quadratic polynomial \(ax^2 + bx + c\), the relationships between the zeroes (\(x_1\) and \(x_2\)) and coefficients are given by:
- **Sum of Zeroes**:
\[
x_1 + x_2 = -\frac{b}{a} = -\frac{-18}{6} = 3
\]
- **Product of Zeroes**:
\[
x_1 \cdot x_2 = \frac{c}{a} = \frac{12}{6} = 2
\]
**Verification**:
- Sum of zeroes: \(2 + 1 = 3\)
- Product of zeroes: \(2 \cdot 1 = 2\)
Both relationships hold true, confirming the consistency of polynomial properties.
Step 1: Finding the Zeroes
To find the zeroes, we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \(a = 6\), \(b = -18\), and \(c = 12\).
- Calculate the discriminant (\(D\)):
\[
D = b^2 - 4ac = (-18)^2 - 4 \cdot 6 \cdot 12 = 324 - 288 = 36
\]
- Now, apply the quadratic formula:
\[
x = \frac{18 \pm \sqrt{36}}{2 \cdot 6} = \frac{18 \pm 6}{12}
\]
- This gives:
- \(x_1 = \frac{24}{12} = 2\)
- \(x_2 = \frac{12}{12} = 1\)
Thus, the zeroes are \(x = 2\) and \(x = 1\).
Step 2: Relationship Between Zeroes and Coefficients
For a quadratic polynomial \(ax^2 + bx + c\), the relationships between the zeroes (\(x_1\) and \(x_2\)) and coefficients are given by:
- **Sum of Zeroes**:
\[
x_1 + x_2 = -\frac{b}{a} = -\frac{-18}{6} = 3
\]
- **Product of Zeroes**:
\[
x_1 \cdot x_2 = \frac{c}{a} = \frac{12}{6} = 2
\]
**Verification**:
- Sum of zeroes: \(2 + 1 = 3\)
- Product of zeroes: \(2 \cdot 1 = 2\)
Both relationships hold true, confirming the consistency of polynomial properties.
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Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.?
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Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.?.
Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find zeroes of a polynomial 6x²-18x+12 and verify the relationship between zeroes and coefficients of variables in the polynomial.?.
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