Questions and Answers
What is the nature of roots for the equation $x^2 - 3x - 10 = 0$?
Which pair of numbers has a sum of 27 and a product of 182?
How many articles were produced if the cost of production of each article is twice the number of articles and the total cost is ₹90?
What are the two consecutive positive integers whose sum of squares equals 365?
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If the altitude of a right triangle is 7 cm less than its base and the hypotenuse is 13 cm, which equation represents this situation?
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What is the value of the discriminant for the equation $3x^2 - 2x + \frac{1}{3} = 0$?
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How many distinct real roots does the equation $3x^2 - 2x + \frac{1}{3} = 0$ have?
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What are the roots of the equation $3x^2 - 2x + \frac{1}{3} = 0$?
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What values of $a$, $b$, and $c$ correspond to the quadratic equation $3x^2 - 2x + \frac{1}{3} = 0$?
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Why is the discriminant important in determining the nature of the roots of a quadratic equation?
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Study Notes
Quadratic Equations and Discriminants
- Standard form of a quadratic equation: ( ax^2 + bx + c = 0 ), where ( a, b, c ) are real numbers, and ( a \neq 0 ).
- Discriminant formula: ( D = b^2 - 4ac ).
- Nature of roots based on discriminant:
- If ( D > 0 ): Two distinct real roots.
- If ( D = 0 ): Two equal real roots.
- If ( D < 0 ): No real roots; complex roots exist.
Example and Applications
- For the equation ( 3x^2 - 2x + \frac{1}{3} = 0 ):
- ( a = 3, b = -2, c = \frac{1}{3} ).
- Discriminant ( D = (-2)^2 - 4 \cdot 3 \cdot \frac{1}{3} = 0 ), indicating two equal real roots.
- Roots found: ( x = \frac{-b}{2a} = \frac{1}{3} ).
Factorization Method for Roots
- Example quadratic ( 2x^2 - 3x + 1 = 0 ):
- One root found by substitution: ( x = 1 ).
- Factorization leads to ( (2x - 1)(x - 1) = 0 ) yielding roots ( x = \frac{1}{2}, 1 ).
Solving Specific Quadratic Equations
- Roots of quadratic equations can be found through factorization:
- Example equations to factor:
- ( x^2 - 3x - 10 = 0 )
- ( 2x^2 + x - 6 = 0 )
- ( \sqrt{2}x^2 + 7x + 5\sqrt{2} = 0 )
- ( 2x^2 - x + \frac{1}{8} = 0 )
- ( 100x^2 - 20x + 1 = 0 )
- Example equations to factor:
Finding Roots Through the Quadratic Formula
- Roots can also be calculated using the formula:
- ( x = \frac{-b \pm \sqrt{D}}{2a} ) where ( D = b^2 - 4ac ).
Additional Problems
- Find two numbers whose sum is 27 and product is 182.
- Find consecutive positive integers such that the sum of their squares is 365.
- Determine sides of a right triangle where one side is shorter than the hypotenuse by 7 cm and hypotenuse is 13 cm.
- Solve for the number of items produced by a pottery industry given total production costs.
Key Takeaways
- Quadratic equations may possess a maximum of two roots, related to their factorization and polynomial behavior.
- Utilize the discriminant to assess the nature of roots quickly.
- Factorization is a valuable method to derive roots when applicable.
- The quadratic formula is a universal tool for finding roots.
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Description
This quiz focuses on determining the discriminant of a given quadratic equation and analyzing the nature of its roots. You will also calculate the roots if they are real, with specific examples to enhance your understanding of quadratic functions.
