Quadratic Equations and Their Graphs - Class 10
Introduction:
A quadratic equation is a second-degree polynomial equation in one variable (usually represented as 'x'). The general form of a quadratic equation is given by: ax^2 + bx + c = 0 where 'a', 'b', and 'c' are constants, and 'a' must not be equal to zero.
Solving Quadratic Equations:
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Factoring Method: If the quadratic equation can be factored, set each factor equal to zero and solve for 'x'.
Example: Solve the equation x^2 - 5x + 6 = 0
Factoring: (x - 2)(x - 3) = 0
Setting each factor equal to zero:
- x - 2 = 0 => x = 2
- x - 3 = 0 => x = 3
The solutions are x = 2 and x = 3.
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Quadratic Formula: If the quadratic equation cannot be easily factored, we can use the quadratic formula to find the solutions.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Example: Solve the equation 2x^2 - 7x + 3 = 0
Using the quadratic formula:
x = [7 ± √(7^2 - 4*2*3)] / 2*2
x = [7 ± √(49 - 24)] / 4
x = [7 ± √25] / 4
x = (7 + 5) / 4 or x = (7 - 5) / 4
The solutions are x = 3 and x = 0.5.
Nature of Roots:
The nature of the roots (solutions) of a quadratic equation depends on the discriminant (D) given by the expression (b^2 - 4ac).
- D > 0: Two distinct real roots. The graph intersects the x-axis at two points.
- D = 0: One real root (repeated root). The graph touches the x-axis at one point (vertex).
- D < 0: No real roots. The graph does not intersect the x-axis (no real solutions).
Graph of Quadratic Equation:
The graph of a quadratic equation is a parabola. It can open upwards or downwards, depending on the coefficient 'a'.
- If 'a' is positive (a > 0), the parabola opens upwards, and it has a minimum point at the vertex.
- If 'a' is negative (a < 0), the parabola opens downwards, and it has a maximum point at the vertex.
Vertex of the Parabola:
The vertex of the parabola is the highest or lowest point on the graph. The x-coordinate of the vertex is given by:
x-coordinate of vertex = -b / 2a
Axis of Symmetry:
The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two equal halves.
Graph Transformations:
- Vertical Translation: Adding or subtracting a constant 'k' to the function shifts the graph up or down.
Example: y = x^2 + 2 shifts the graph of y = x^2 two units upwards.
- Horizontal Translation: Replacing 'x' with '(x - h)' in the function shifts the graph left or right.
Example: y = (x - 3)^2 shifts the graph of y = x^2 three units to the right.
- Vertical Stretch/Compression: Multiplying the function by a constant 'k' stretches or compresses the graph vertically.
Example: y = 2x^2 stretches the graph of y = x^2 vertically by a factor of 2.
- Reflection: Replacing 'x' with '(-x)' or 'y' with '(-y)' in the function reflects the graph across the x-axis or y-axis, respectively.
Example: y = -x^2 reflects the graph of y = x^2 across the x-axis.
Finding Maximum or Minimum Value:
For a quadratic equation in the form y = ax^2 + bx + c:
- If 'a' is positive (a > 0), the parabola opens upwards, and the vertex represents the minimum value of the function.
- If 'a' is negative (a < 0), the parabola opens downwards, and the vertex represents the maximum value of the function.
Applications of Quadratic Equations:
Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. Some common applications include finding the maximum or minimum values of functions, calculating projectile motion, determining profit-maximizing levels in business, and designing computer algorithms.
Summary:
- Quadratic equations are second-degree polynomial equations in one variable. - They can be solved using factoring or the quadratic formula. - The nature of roots depends on the discriminant (D = b^2 - 4ac). - The graph of a quadratic equation is a parabola, which can open upwards or downwards. - The vertex represents the highest or lowest point on the graph, and the axis of symmetry divides the parabola into two equal halves. - Quadratic equations have real-world applications in various fields.
Quadratic Equation Solver
Enter the coefficients of the quadratic equation (ax^2 + bx + c = 0) below: