Each expression represents two numbers. Evaluate the expressions and find the two numbers.
16.2
Activity
Pesky Equations
Choose one equation to solve, either by rewriting it in factored form or by completing the square. Be prepared to explain your choice of method.
16.3
Activity
Meet the Quadratic Formula
Here is a formula called the quadratic formula.
The formula can be used to find the solutions to any quadratic equation in the form of , where , , and are numbers and is not 0.
This example shows how it is used to solve , in which , , and .
Here are some quadratic equations and their solutions. Use the quadratic formula to show that the solutions are correct.
. The solutions are and .
. The solutions are and .
. The solutions are .
. The solutions are .
. The solution is .
. The solutions are and .
Student Lesson Summary
We have learned a couple of methods for solving quadratic equations algebraically:
By rewriting the equation as so that one side is 0 and the other side is in factored form, then using the zero product property
By completing the square
Some equations can be solved quickly with one of these methods, but many cannot. Here is an example: . The expression on the left cannot be rewritten in factored form with rational coefficients. Because the coefficient of the squared term is not a perfect square, and the coefficient of the linear term is an odd number, completing the square would be inconvenient and would result in a perfect square with fractions.
The quadratic formula can be used to find the solutions to any quadratic equation, including those that are tricky to solve with other methods.
For an equation of the form , where , , and are numbers and , the solutions are given by:
For the equation , we see that , , and . Let’s solve it!
A calculator gives approximate solutions of 0.84 and -0.24 for and .
We can also use the formula for simpler equations like , but it may not be the most efficient way. If the quadratic expression can be easily rewritten in factored form or made into a perfect square, those methods may be preferable. For example, rewriting as immediately tells us that the solutions are 1 and 8.
Glossary
quadratic formula
The quadratic formula is and gives the solutions of the quadratic equation , where , , and are constants and .
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