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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
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.
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.
Quadratic functions are especially easy to describe: if y = f(x) = ax2 + bx + c, then. the y-intercept is y = c. the x-intercepts are two if b2 − 4ac > 0: b2 they reduce to one if b2 − 4ac = 0: −. 2a. there are none if b2 − 4ac < 0. if a > 0 the function has a minimum for x = −. b2 is −. .
Quadratic Equation. 1. The definition and main notations. The general form of quadratic equation is. ax2 + bx + c = 0. Where a, b, c are constants. The function. f (x) = ax2 + bx + c. describes a parabola, which looks like this graph below.
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.
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.
