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Quadratic equations are equations with the form
The expression is called a quadratic expression, because the highest power of any of the terms is 2. There are four methods for solving quadratic equations by hand:
The results of these methods will always be the roots of the function.
Formula
The quadratic formula can be used with all quadratic expressions. The roots are
Quadratic equations can have no solutions, one solution or two solutions.
no real solutions,
one real solution,
two real solutions.
Example 1
Solve the equation .
First, you have to put all the nonzero terms on one side of the equal sign to get alone on the other side:
Next, you use the quadratic formula with , and :
Set up the expressions with both the positive square root and the negative square root:
That means the solutions are and .
Rule
When , the expression looks like this:
Example 2
Solve the equation .
You factorize by taking out of the parentheses:
Then you set up an equation for each factor:
That will give you and .
Rule
When , the expression looks like this:
Example 3
Rule
You have . When you solve by inspection, you follow these two rules:
Here, and are the roots of the function, making them the solutions to the equation.
Example 4
Solve the equation .
You can see that and . You need to find values for and that can get you the solution to the equation. There are multiple combinations of factors that give you the product . Here are some of them:
As all of these products are , they are all candidates for the solution. For each of these products, we want a negative difference between the factors, because you are looking for the answer , a negative number. That means we set the differences up like this:
That shows you that and fits with the equation. According to the formula, that means
because you have to change the sign to find the solution.