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 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.) These 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. Inspect this graph of
to solve this inequality:
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. These three possibilities for any real number -- positive, negative, or 0 -- is called the Law of Trichotomy.
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. Inspect the graph of
The quadratic will be positive -- above the 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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