10 ## QUADRATICS## Polynomials of the 2nd degreeSolving the quadratic equation by factoring The sum and product of the roots QUADRATIC IS ANOTHER NAME for a polynomial of the 2nd degree. 2 is the highest exponent. 1. A polynomial function of the 2nd degree has what form?
2. A quadratic equation has what form?
3. What do we mean by a root of a quadratic? A solution to the quadratic equation. 4. A quadratic always has how many roots? Two, real or complex. 5. The graph of a quadratic is always the form called -- ? A parabola. 6. What are the three methods for solving a quadratic equation, 1. Factoring. 2. Completing the square. 3. The quadratic formula. We begin with the method of factoring. In the next Topic, we will present both Completing the square and The quadratic formula. 7. If a product of factors is 0 -- if Either Example 1. Solution by factoring.
Therefore, the roots are −1 and 3. (See Lesson 37 of Algebra.) They are the The In every polynomial, the Example 2. A double root
At a double root, the graph does not cross the A double root occurs when the quadratic is a perfect square trinomial: Example 3. How many real roots, i.e. roots that are real numbers, has the quadratic of each graph? Graph b) has no real roots. It has no Graph c) has two real roots. But they are a double root. Example 4. Quadratic inequality. Solve this inequality:
To do it, inspect the graph of
The graph is negative between the roots, which are −1 and 5. The solution to the inequality is −1 < We can also observe that the quadratic will have positive values -- the graph will be above the
While the quadratic will have the value 0 at the roots. We have now covered the three possibilities: That quadratic is It is It is These three possiblities, which are true for any real number, has the fancy name of the Law of Trichotomy. Any number must be either equal to, less than, or greater than 0. The Law of Trichotomy also takes this form: For any real numbers We must be able to know, though, which of those possibilities is true. For any two numbers, we must be able to know their relative order. That is inherent in the meaning of a "number."
Problem 1. Sketch the graph of To see the answer, pass your mouse over the colored area.
Problem 2. Sketch the graph of
Problem 3. a) To solve this quadratic inequality—
—inspect the graph of
The quadratic will be positive -- above the b) Solve this quadratic inequality:
−3 < The quadratic will be negative between the roots. Problem 4. A quadratic has the following roots. Write each quadratic as a product of linear factors. a) 3, 4
( b) −3, −4
( c) − d) 3 + , 3 −
( The sum and product of the roots
Theorem. The sum of the roots is the the product of the roots is the constant term. That is, if
and the roots are
For if the roots are
The coefficient of Example 5. Construct the quadratic whose roots are 2 and 3. The sum of the roots is the Example 6. Construct the quadratic whose roots are 2 + , 2 − . The quadratic therefore is
Example 7. Construct the quadratic whose roots are 2 + 3 The quadratic with those roots is
Problem 5. Construct the quadratic whose roots are −3, 4.
The sum of the roots is 1. Their product is −12. Therefore, the quadratic is Problem 6. Construct the quadratic whose roots are 3 + , 3 − .
The sum of the roots is 6. Their product is 9 − 3 = 6.
Problem 7. Construct the quadratic whose roots are 2 +
The sum of the roots is 4. Their product is 4 − ( * More generally, for any coefficient of
and the roots are
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