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A quadratic equation takes the form ax2 +bx+c = 0 where a, b and c are numbers. The number a cannot be zero. In this unit we will look at how to solve quadratic equations using four methods: •solution by factorisation •solution by completing the square •solution using a formula •solution using graphs
This topic introduces quadratic functions, their graphs and their important characteristics. Quadratic functions are widely used in mathematics and statistics. They are found in applied and theoretical mathematics, and are used to model non-linear relationships between variables in statistics.
To solve a quadratic equation by graphing, first write the equation in standard form, ax2 1 bx 1 c 5 0. Then graph the related function. y 5 ax2 1 bx 1 c. The x-intercepts of the graph are the solutions, or roots, of ax2 1 bx 1 c 5 0.
An algebraic expression can change signs only at the x-values that make the expression zero or undefined. The zeros and undefined values make up the critical numbers of the expression, and they are used to determine the test intervals in solving quadratic and rational inequalities.
Quadratic formula: Solution to ax2 + bx+ c = 0 is x = 2b p b 4ac 2a. There are three possibilities for the solution, based on the sign of the quantity b2 4ac.
QUADRATIC FUNCTIONS. PROTOTYPE: (x) = ax2 + bx + c: The leading coe±cient a 6= 0 is called the shape parameter. SHAPE-VERTEX FORMULA. One can write any quadratic function (1) as. f (x) = a(x ¡ h)2 + k; (Shape-Vertex Formula) b b2. where h = ¡ and k = f (h) = c ¡ . 2a 4a. EXAMPLE 1. f (x) = 2x2 ¡ 8x + 5.
We will derive a transformation form for a general quadratic function, an equation that identifies the vertex and axis of symmetry of the graph, but to graph any particular quadratic, you may not need all of the steps.
