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Questions and Answers
Explain the process of solving a quadratic equation using the quadratic formula.
The quadratic formula states that for the equation ax^2 + bx + c = 0, the solutions for x are given by x = (-b ± √(b^2 - 4ac)) / (2a). To solve a quadratic equation using the quadratic formula, first identify the values of a, b, and c in the equation, then substitute these values into the formula and solve for x using the ± symbol to represent both the positive and negative roots.
How does the method of completing the square help in solving a quadratic equation?
The method of completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in the form (x + p)^2 = q. By doing this, the equation can be easily solved for x by taking the square root of both sides. This method is particularly useful when the quadratic equation cannot be easily factored or when solving for the roots using other methods becomes complicated.
What is the discriminant of a quadratic equation and how is it used to determine the nature of the roots?
The discriminant of a quadratic equation, given by the expression b^2 - 4ac, is used to determine the nature of the roots. If the discriminant is greater than 0, the equation has two distinct real roots. If the discriminant is equal to 0, the equation has two equal real roots. If the discriminant is less than 0, the equation has no real roots, but two complex roots.
Explain the steps to solve the quadratic equation 3x^2 - 5x - 2 = 0 using the method of factorization.
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If a quadratic equation has one root as 3, what is the value of the other root if the sum of the roots is 5?
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What is the sum of the roots of the quadratic equation 4x^2 - 7x + 2 = 0 and what are the roots?
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Study Notes
Solving Quadratic Equations
- The quadratic formula is a method for solving quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants.
- The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where ± indicates two possible solutions.
Method of Completing the Square
- Completing the square is a method for solving quadratic equations by rearranging the equation to have a perfect square on one side and a constant on the other.
- The method involves adding and subtracting a value to both sides of the equation to create a perfect square trinomial.
Discriminant of a Quadratic Equation
- The discriminant of a quadratic equation is the expression b^2 - 4ac, where a, b, and c are the coefficients of the equation.
- The discriminant is used to determine the nature of the roots:
- If the discriminant is positive, the equation has two distinct real roots.
- If the discriminant is zero, the equation has one repeated real root.
- If the discriminant is negative, the equation has no real roots (but has two complex roots).
Solving Quadratic Equations using Factorization
- The method of factorization involves expressing the quadratic equation as a product of two binomials.
- The equation 3x^2 - 5x - 2 = 0 can be solved by factorization as follows:
- Factor out the greatest common factor (GCF) of the coefficients.
- Express the equation as a product of two binomials.
- Equate each factor to zero and solve for x.
Properties of Quadratic Equations
- If one root of a quadratic equation is 3, and the sum of the roots is 5, then the other root is 5 - 3 = 2.
- The sum of the roots of the quadratic equation 4x^2 - 7x + 2 = 0 is -(-7) / 4 = 7/4.
- The roots of the equation 4x^2 - 7x + 2 = 0 can be found using the quadratic formula or other methods.
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Description
Test your knowledge of quadratic equations with this class 10 ICSE quiz, which covers miscellaneous problems to help reinforce your understanding.
