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If a and ẞ are roots of the quadratic equation 2x ^ 2 - 4x - 3 + 0 find the quadratic equation whose roots are alpha ^ 2 and beta ^ 2 And find the value of alpha ^ 2 - beta ^ 2?
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If a and ẞ are roots of the quadratic equation 2x ^ 2 - 4x - 3 + 0 fin...
Given Quadratic Equation
The given quadratic equation is:
2x² - 4x - 3 = 0
Finding Roots (Alpha and Beta)
To find the roots, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Here, a = 2, b = -4, and c = -3.
- Discriminant: D = b² - 4ac = (-4)² - 4(2)(-3) = 16 + 24 = 40
- Roots:
- Alpha (α) = [4 + √40]/4 = (4 + 2√10)/4 = 1 + √10/2
- Beta (β) = [4 - √40]/4 = (4 - 2√10)/4 = 1 - √10/2
New Quadratic Equation with Roots Alpha² and Beta²
To find the new quadratic equation whose roots are α² and β², we use the relationships:
- Sum of roots (α² + β²) = (α + β)² - 2αβ
- Product of roots (α²β²) = (αβ)²
Using Vieta's formulas from the original equation:
- α + β = -(-4)/2 = 2
- αβ = -3/2
Now, calculate the new values:
- α² + β² = (2)² - 2(-3/2) = 4 + 3 = 7
- α²β² = (-3/2)² = 9/4
Formulating the New Equation
The quadratic equation is formed as:
x² - (α² + β²)x + α²β² = 0
Substituting values:
x² - 7x + 9/4 = 0
Multiplying by 4 to eliminate fractions:
4x² - 28x + 9 = 0
Finding Alpha² - Beta²
Using the difference of squares formula:
α² - β² = (α + β)(α - β)
We already know α + β = 2.
Now, to find α - β:
α - β = √40 / 4 = √10 / 2
Thus:
α² - β² = 2 * (√10/2) = √10
Final Results
- New Quadratic Equation: 4x² - 28x + 9 = 0
- Value of α² - β²: √10
The given quadratic equation is:
2x² - 4x - 3 = 0
Finding Roots (Alpha and Beta)
To find the roots, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
Here, a = 2, b = -4, and c = -3.
- Discriminant: D = b² - 4ac = (-4)² - 4(2)(-3) = 16 + 24 = 40
- Roots:
- Alpha (α) = [4 + √40]/4 = (4 + 2√10)/4 = 1 + √10/2
- Beta (β) = [4 - √40]/4 = (4 - 2√10)/4 = 1 - √10/2
New Quadratic Equation with Roots Alpha² and Beta²
To find the new quadratic equation whose roots are α² and β², we use the relationships:
- Sum of roots (α² + β²) = (α + β)² - 2αβ
- Product of roots (α²β²) = (αβ)²
Using Vieta's formulas from the original equation:
- α + β = -(-4)/2 = 2
- αβ = -3/2
Now, calculate the new values:
- α² + β² = (2)² - 2(-3/2) = 4 + 3 = 7
- α²β² = (-3/2)² = 9/4
Formulating the New Equation
The quadratic equation is formed as:
x² - (α² + β²)x + α²β² = 0
Substituting values:
x² - 7x + 9/4 = 0
Multiplying by 4 to eliminate fractions:
4x² - 28x + 9 = 0
Finding Alpha² - Beta²
Using the difference of squares formula:
α² - β² = (α + β)(α - β)
We already know α + β = 2.
Now, to find α - β:
α - β = √40 / 4 = √10 / 2
Thus:
α² - β² = 2 * (√10/2) = √10
Final Results
- New Quadratic Equation: 4x² - 28x + 9 = 0
- Value of α² - β²: √10
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If a and ẞ are roots of the quadratic equation 2x ^ 2 - 4x - 3 + 0 find the quadratic equation whose roots are alpha ^ 2 and beta ^ 2 And find the value of alpha ^ 2 - beta ^ 2?
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