Let us consider the following quadratic function: f(x) = -x^{2} + 15x -20
The solutions of the quadratic equation -x^{2} + 15x -20 = 0 correspond to the zeros or the roots of the function f(x) = -x^{2} + 15x -20.
These are the points where the graph of f(x) cuts the x-axis. The graph cuts the Y-axis at -20.
The Discriminant and Roots of the Quadratic Equation -x^{2} + 15x -20 = 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 = 15, c = -20.
The discriminant D = b^{2} - 4ac = 15^{2} - 4 * (-1) * (-20) = 145.0
The roots of the equation are (-b - √D)/2a and (-b + √D)/2a
= (-(15) - √145.0)/(2(-1)) and (-(15) + √145.0)/(2(-1))
= 13.521 and 1.479
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 < 13.521) and (x > 1.479) satisfy the inequality -x^{2} + 15x -20 < 0 Graph of y = f(x) = -x^{2} + 15x -20
Geometric and Graphical interpretation: Curve SketchingThe function f(x) = -x^{2} + 15x -20 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 maxima point and opens downwards. 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) = (-(15))/(2*-1) = 7.5, and the y-coordinate of the vertex is 36.25 which we obtain by substituting the value x = 7.5 in -x^{2} + 15x -20. This is the maxima 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} + 15x -20 = 0 correspond to the zeros or the roots of the function f(x) = -x^{2} + 15x -20. 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 < 13.521) and (x > 1.479) satisfy the inequality -x^{2} + 15x -20 < 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. |
The Learning Point > Mathematics > Quadratic Equations, Cubic and Higher Order Equations : Plots, Factorization, Formulas >