Problem 17
Question
Let \(N\) be the number of quadratic equations with coefficients \(\\{0,1,2, \ldots, 9\\}\) such that zero is a solution of each equation. [Kerala PET - 2003] Then the value of \(N\) is (a) Infinite (b) \(2^{9}\) (c) 90 (d) 900
Step-by-Step Solution
Verified Answer
The value of \( N \) is 99.
1Step 1: Understand the Form of the Quadratic Equation
The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). Given that zero is a solution, it must satisfy the equation. Substituting \( x = 0 \) gives \( c = 0 \). Therefore, the equation simplifies to \( ax^2 + bx = 0 \).
2Step 2: Determine the Conditions for Non-zero Coefficients
For the equation \( ax^2 + bx = 0 \) to be valid, at least one of \( a \) or \( b \) must be non-zero (otherwise the equation would be identically zero). Both \( a \) and \( b \) can independently take any value from the set \( \{0, 1, 2, \ldots, 9\} \).
3Step 3: Calculate Total Combinations for Non-zero Equation
Since \( a \) and \( b \) each have 10 possible values (0 through 9), and both cannot simultaneously be zero, calculate the total number of combinations: \( 10 \times 10 = 100 \). However, since \( (a, b) = (0, 0) \) is not allowed, subtract 1 to get \( 100 - 1 = 99 \) valid combinations.
4Step 4: Conclusion
Each combination of \( a \) and \( b \) forms a unique quadratic equation with \( c = 0 \) satisfying the condition that zero is a solution. Thus, there are 99 quadratic equations possible under these conditions.
Key Concepts
Coefficient valuesSolution of equationsAlgebraic expressions
Coefficient values
In a quadratic equation, the coefficients determine the shape and position of its graph. The general form of a quadratic equation is described by \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients that must be carefully chosen to represent various scenarios.
In the specific scenario given in the exercise, where zero is a solution, the constant term, \( c \), becomes zero. This is because substituting zero for \( x \) simplifies the equation to \( ax^2 + bx = 0 \). So, this problem simplifies further because the coefficient values \( a \) and \( b \) could be any numbers from the set \( \{0,1,2,\ldots, 9\} \), as long as they aren't both zero at the same time. This ensures that a valid quadratic equation can be constructed.
By understanding the role of coefficients, especially in this case where \( c \) is naturally zero to satisfy the condition of zero being a solution, one can explore a huge variety of quadratic forms by just altering \( a \) and \( b \). This provides insight into how these values intricately control the behavior of quadratic expressions.
In the specific scenario given in the exercise, where zero is a solution, the constant term, \( c \), becomes zero. This is because substituting zero for \( x \) simplifies the equation to \( ax^2 + bx = 0 \). So, this problem simplifies further because the coefficient values \( a \) and \( b \) could be any numbers from the set \( \{0,1,2,\ldots, 9\} \), as long as they aren't both zero at the same time. This ensures that a valid quadratic equation can be constructed.
By understanding the role of coefficients, especially in this case where \( c \) is naturally zero to satisfy the condition of zero being a solution, one can explore a huge variety of quadratic forms by just altering \( a \) and \( b \). This provides insight into how these values intricately control the behavior of quadratic expressions.
Solution of equations
For students learning about quadratic equations, identifying a solution is a critical step. In the context of this exercise, the fact that zero is a solution has great importance. Typically, a solution is a value of \( x \) that makes the equation true. Here, it means that the expression \( ax^2 + bx + c = 0 \) holds true when \( x = 0 \).
By setting \( x = 0 \), we immediately know that \( c \) must be zero since any non-zero \( c \) would result in a contradiction (the equation would not equal zero). The presence of zero as a solution then simplifies our equation to \( ax^2 + bx = 0 \). The task then becomes to ensure the equation is valid for non-trivial cases, i.e., not just when \( a = 0 \) and \( b = 0 \) simultaneously.
Finding solutions often involves substituting values or factoring, but in this problem, the provided condition \( x = 0 \) directly dictates the necessary structure of the equation. This deductive reasoning demonstrates an applied understanding of how solutions are interconnected with the equation's representation and its coefficients.
By setting \( x = 0 \), we immediately know that \( c \) must be zero since any non-zero \( c \) would result in a contradiction (the equation would not equal zero). The presence of zero as a solution then simplifies our equation to \( ax^2 + bx = 0 \). The task then becomes to ensure the equation is valid for non-trivial cases, i.e., not just when \( a = 0 \) and \( b = 0 \) simultaneously.
Finding solutions often involves substituting values or factoring, but in this problem, the provided condition \( x = 0 \) directly dictates the necessary structure of the equation. This deductive reasoning demonstrates an applied understanding of how solutions are interconnected with the equation's representation and its coefficients.
Algebraic expressions
Algebraic expressions form the foundation of solving equations, especially quadratic ones. Here, expressions are used to set up the equation where the goal is to find valid configurations such that zero is a solution. The expression \( ax^2 + bx \) is what remains after reducing the initial quadratic form \( ax^2 + bx + c \) by setting \( c = 0 \).
Algebraic expressions can be manipulated in various ways, such as factoring or expanding. In this problem, the expression \( ax^2 + bx \) can be factored as \( x(ax+b) = 0 \). This shows that either \( x = 0 \) or \( ax + b = 0 \), aligning with the requirement that zero is inherently a solution or other arrangements of \( a \) and \( b \) ensure the equation holds.
Understanding how to manipulate these expressions using algebraic techniques is key in deriving different conditions under the given constraints. This specific exercise shows the power of simplification and logical deduction in enhancing one's ability to manage algebraic forms, turning them from abstract concepts into solvable representations.
Algebraic expressions can be manipulated in various ways, such as factoring or expanding. In this problem, the expression \( ax^2 + bx \) can be factored as \( x(ax+b) = 0 \). This shows that either \( x = 0 \) or \( ax + b = 0 \), aligning with the requirement that zero is inherently a solution or other arrangements of \( a \) and \( b \) ensure the equation holds.
Understanding how to manipulate these expressions using algebraic techniques is key in deriving different conditions under the given constraints. This specific exercise shows the power of simplification and logical deduction in enhancing one's ability to manage algebraic forms, turning them from abstract concepts into solvable representations.
Other exercises in this chapter
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