Surprisingly, when mathematics is employed to solve complicated and important real world problems, quadratic equations very often make an appearance as part of the overall solution. Both in senior mathematics and in tertiary and engineering mathematics, students will need to be able to solve quadratic equations with confidence and speed. While quadratic equations do not arise so obviously in everyday life, they are equally important and will frequently turn up in many areas of mathematics when more sophisticated problems are encountered. In this module we will develop a number of methods of dealing with these important types of equations. The rearrangements we used for linear equations are helpful but they are not sufficient to solve a quadratic equation. We keep rearranging the equation so that all the terms involving the unknown are on one side of the equation and all the other terms to the other side. The essential idea for solving a linear equation is to isolate the unknown. The equation = is also a quadratic equation. Thus, for example, 2 x 2 − 3 = 9, x 2 − 5 x + 6 = 0, and − 4 x = 2 x − 1 are all examples of quadratic equations. Roughly speaking, quadratic equations involve the square of the unknown. Such equations arise very naturally when solving elementary everyday problems.Ī linear equation involves the unknown quantity occurring to the first power, thus, Any other quadratic equation is best solved by using the Quadratic Formula.In the module, Linear Equations we saw how to solve various types of linear equations. If the equation fits the form ax 2 = k or a( x − h) 2 = k, it can easily be solved by using the Square Root Property. If the quadratic factors easily, this method is very quick. How to identify the most appropriate method to solve a quadratic equation.if b 2 − 4 ac if b 2 − 4 ac = 0, the equation has 1 real solution.If b 2 − 4 ac > 0, the equation has 2 real solutions.For a quadratic equation of the form ax 2 + bx + c = 0,.Using the Discriminant, b 2 − 4 ac, to Determine the Number and Type of Solutions of a Quadratic Equation.Then substitute in the values of a, b, c. Write the quadratic equation in standard form, ax 2 + bx + c = 0. How to solve a quadratic equation using the Quadratic Formula.We start with the standard form of a quadratic equation and solve it for x by completing the square. Now we will go through the steps of completing the square using the general form of a quadratic equation to solve a quadratic equation for x. We have already seen how to solve a formula for a specific variable ‘in general’, so that we would do the algebraic steps only once, and then use the new formula to find the value of the specific variable. In this section we will derive and use a formula to find the solution of a quadratic equation. Mathematicians look for patterns when they do things over and over in order to make their work easier. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes’. When we solved quadratic equations in the last section by completing the square, we took the same steps every time. Solve Quadratic Equations Using the Quadratic Formula
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