37 ## QUADRATIC EQUATIONSThe standard form of a quadratic equation Proof of the quadratic formula The graph of A QUADRATIC is a polynomial whose highest exponent is 2.
The coefficient of Question 1. What is the standard form of a
The quadratic is on the left. 0 is on the right. Question 2. What do we mean by a root of a quadratic? A solution to the quadratic equation. For example, the roots of this quadratic --
-- are the solutions to
To find the roots, we can factor that quadratic as ( Now, if
−4 and 2 are the solutions to the quadratic equation. They are the roots of that quadratic. Conversely, if the roots are ( A root of a quadratic is also called a zero. Because, as we will see, at each root the value of the graph is 0. (See Topic 7 of Precalculus, Question 2.) Question 3. How many roots has a quadratic?
Always two. Because a quadratic (with leading coefficient 1, at least) can always be factored as (
In other words, when the leading coefficient is 1, the root has the opposite sign of the number in the factor. −
Problem 1. If a quadratic can be factored as ( To see the answer, pass your mouse over the colored area. −3 or 1. We say "or," because Question 4. What do we mean by a double root?
The two roots are equal. The factors will be ( For example, this quadratic
can be factored as ( If When will a quadratic have a double root? When the quadratic is a perfect square trinomial.
Example 1. Solve for
2 Now, it is easy to see that the second factor will be 0 when As for the value of
The solutions are:
Problem 2. How is it possible that the product of two factors
Either Solution by factoring Problem 3. Find the roots of each quadratic by factoring.
Again, we use the conjunction "or," because
Example 2.
Those are the two roots. Problem 4. Find the roots of each quadratic.
Example 3.
However, if the form is the difference of two squares --
-- then we can factor it as: ( The roots are ±4. In fact, if the quadratic is
then we could factor it as: ( so that the roots are ±. Problem 5. Find the roots of each quadratic.
Example 4. Solve this quadratic equation:
And so an equation is solved when
Problem 6. Solve each equation for
Example 5. Solve this equation
Next, we can get rid of the fraction by multiplying both sides by 2. Again, 0 will not change.
Problem 7. Solve for
Section 2: Completing the square Please make a donation to keep TheMathPage online. Copyright © 2015 Lawrence Spector Questions or comments? E-mail: themathpage@nyc.rr.com |