Discuss three symbolic methods for solving a quadratic equation. Make up a quadratic equation an

Discuss three symbolic methods for solving a quadratic...

Key Concepts

Quadratic Formula

The quadratic formula is a generalized method for solving any quadratic equation. It is derived from the process of completing the square and is given by x = (-b ± ?(b² - 4ac))/(2a). This formula directly provides the solutions of a quadratic equation and is especially valuable when the equation does not factor neatly.

Completing the Square

Completing the square is a technique that transforms the quadratic equation into a perfect square trinomial plus a constant. This is accomplished by manipulating the equation to express it in the form (x + d)² = e, where d and e are derived from the coefficients of the equation. This method is particularly useful for deriving the quadratic formula and for understanding the geometric interpretation of quadratics.

Factorization

Factorization involves rewriting the quadratic equation as a product of two simpler binomial expressions. Once the equation is factored, the Zero-Product Property is used to set each binomial equal to zero, which then provides the solutions for the variable. This method is effective when the quadratic readily factors into integers or simple expressions.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0 (with a ? 0). It is a fundamental concept in algebra that models various phenomena and is solved to find the values of x that satisfy the equation.

Transcript

00:02 Okay, so whenever you are asked to solve symbolically or when you are given no specific instructions for the method to use, it is always appropriate to use a symbolic method.

00:18 So we have three times.

00:20 It's either you use the factoring method.

00:23 So we have the factoring method.

00:26 And the second one is completing the squares.

00:31 So completing the squares.

00:36 And the last one, is using the quadratic formula.

00:50 So what you should know is that not all quadratics can be factored.

00:54 So the other two methods are needed.

00:56 So these two methods are really needed, since we cannot use the factoring method for complex quadratic equations.

01:06 So what you should also know is that when you use these two methods, you should get the same solution no matter which method you use.

01:14 So they should all arrive at the same answer.

01:17 So now let's consider the quadratic equation.

01:20 So this equation, we have x squared minus 3x plus 2.

01:28 So first we are going to use the factoring method, so one, which is f.

01:35 So in the factoring method or in quadratic equations, we know the general form is a x squared plus bx plus c is equal to 0.

01:49 So in this instance, we know a is equal to positive 1.

01:55 We have b equal to negative 3, and we have c to be equal to positive 2 over here.

02:03 So in the fracturing method, this is how it works.

02:08 So we need to find two numbers.

02:12 When you multiply, you are going to get a, c.

02:15 And when you add, you are going to get b.

02:18 So in this situation, since we know a is c, positive 1 and c is negative positive 2.

02:29 When you multiply 1 by 2, you're going to get 2 and we know b here is negative 3.

02:36 So we want two numbers that when you multiply, you're going to get ac which is 2 in this case and when you add the 2 numbers, you are going to get negative 3, which is b in this situation.

02:50 So since our equation is x squared minus 3x plus 2 the two numbers that i chose is negative 2 and negative 1 so when we take negative 2 multiplying negative 1 we are going to get positive 2 and when we add the two numbers which is negative 2 plus negative 1 we are going to get negative 3 so we are right in this situation so now let's proceed so we have x squared minus 2 so we are going to factor these two numbers into our equation.

03:28 So these two numbers, what they are actually going to do is we are going to replace our b x over here.

03:37 So in this situation, negative 3x over here.

03:44 So we are going to get two times x.

03:48 Then the next one is negative one.

03:49 So we have negative x.

03:51 Then we bring our c and i'll see as plus two.

03:57 So we have plus two.

03:57 So we have plus 2 over here equal to 0.

04:01 So from here we can group the times this and this and this and that.

04:10 So we are going to get x.

04:12 We can factor x out from here to get x minus 2.

04:17 And here too, we can factor a negative 1 out to get x minus 1 to be equal to 0.