Definition: A quadratic function is one of the form f(x) = ax2 + bx + c, where a, b, and c are numbers with a not equal to zero. When you solve a quadratic function you are looking for the roots. You can solve a quadratic function in several different ways:
By Factoring: After factoring a quadratic function, the zero factor principal allows us to set each factor equal to zero and solve for x.
For example, f(x) = (x + 2)(x – 7) tells us x + 2 = 0, x = -2 and x – 7 = 0, x = 7.
By Taking Roots: By isolating the squared term and square rooting both sides, we can solve some quadratic functions. For example, given f(x) = x² – 9 we can conclude x² – 9 = 0, x² = 9, x = ±3
By Completing the Square: This method involves rearranging the quadratic so the left side is a perfect square trinomial.
By Using the Quadratic Formula: Starting with a quadratic in standard form, ax² + bx + c, values a, b, and c can be plugged into the quadratic formula:
By Graphing: When graphed, the roots of a quadratic function are the x-intercepts of the graph.
Example: Solve the following quadratic function: f(x) = 3x² + 7x – 12
Step 1: Determine the best method of solving. Because there are no factors of a * c (3*12) whose sum is b (7), we can use the quadratic formula.
Step 2: Plug the given values into the formula. Our quadratic is already in standard form, so we know a = 3, b = 7, c = -12
Step 3: Simplify to find approximate solutions:
x = -3.482, x = 1.148
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