Let us consider the following quadratic function: f(x) = x^{2} -6x -30
The solutions of the quadratic equation x^{2} -6x -30 = 0 correspond to the zeros or the roots of the function f(x) = x^{2} -6x -30.
These are the points where the graph of f(x) cuts the x-axis. The graph cuts the Y-axis at -30.
The Discriminant and Roots of the Quadratic Equation x^{2} -6x -30 = 0The standard form of a quadratic equation is ax^{2} + bx + c = 0, where "a" does not equal 0. Note that if a = 0, the x^{2} term would disappear and the equation would be linear.
Looking at the given quadratic function a = 1, b = -6, c = -30.
The discriminant D = b^{2} - 4ac = -6^{2} - 4 * (1) * (-30) = 156.0
The roots of the equation are (-b - √D)/2a and (-b + √D)/2a
= (-(-6) - √156.0)/(2(1)) and (-(-6) + √156.0)/(2(1))
= -3.245 and 9.245
The discriminant is positive. Hence, the roots are real and unequal. The quadratic curve cuts the X axis at two distinct points. The value of 'x' in the intervals (x < -3.245) and (x > 9.245) satisfy the inequality x^{2} -6x -30 > 0 Graph of y = f(x) = x^{2} -6x -30
Geometric and Graphical interpretation: Curve SketchingThe function f(x) = x^{2} -6x -30 is the quadratic function. The graph of any quadratic function has the same general shape. This shape is called a parabola. The location and size of the parabola, and how it opens, depend on the values of coefficients in the function. This parabola has a minima point and opens upwards. The graph of the parabola is symmetric with respect to the vertical line passing through the vertex. The x-coordinate of the vertex will be located at x = (-b/2a) = (-(-6))/(2*1) = 3.0, and the y-coordinate of the vertex is -39.0 which we obtain by substituting the value x = 3.0 in x^{2} -6x -30. This is the minima value attained by the quadratic function f(x). The derivative of the function is 0 at this point. This point is a turning point or a stationary point.
The solutions of the quadratic equation x^{2} -6x -30 = 0 correspond to the zeros or the roots of the function f(x) = x^{2} -6x -30. As shown in the figure, when the coefficients a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. The discriminant is positive. Hence, the roots are real and unequal. The quadratic curve cuts the X axis at two distinct points.
The value of 'x' in the intervals (x < -3.245) and (x > 9.245) satisfy the inequality x^{2} -6x -30 > 0 . The vertex is the lowest point on the parabola if the parabola opens upward (coefficient a > 0) and is the highest point on the parabola if the parabola opens downward (coefficient a < 0) Check the plot of another quadratic curve here. Here's another quadratic curve here. Many of these concepts are a part of the Grade 9,10,11,12 (High School) Mathematics syllabus of the UK GCSE curriculum, Common Core Standards in the US, ICSE/CBSE/SSC syllabus in Indian high schools. You may check out our free and printable worksheets for Common Core and GCSE. |
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