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37 QUADRATIC EQUATIONS Proof of the quadratic formula A QUADRATIC is a polynomial whose highest exponent is 2. Question 1. What is the standard form of a quadratic equation? To see the answer, pass your mouse over the colored area. ax² + bx + c = 0 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, this quadratic x² + 2x − 8 can be factored as (x + 4)(x − 2). Now, if x = −4, then the first factor will be 0. While if x = 2, the second factor will be 0. But if any factor is 0, then the entire product will be 0. That is, if x = −4 or 2, then x² + 2x − 8 = 0. Therefore, −4 and 2 are the roots of that quadratic. They are the solutions to the quadratic equation. A root of a quadratic is also called a zero. Because, as we will see, at those values of x, the graph has the value 0. Question 3. How many roots has a quadratic? Always two. Question 4. What do we mean by a double root? The two roots are equal. For example, this quadratic x² − 10x + 25 can be factored as (x − 5)(x − 5). If x = 5, then each factor will be 0, and therefore the quadratic will be 0. 5 is called a double root. A quadratic will have a double root if the quadratic is a perfect square trinomial. Problem 1. If either a = 0 or b = 0, then what can you conclude about ab ? ab = 0 Solution by factoring Problem 2. Find the roots of each quadratic by factoring.
Notice that we use the conjunction "or," because x takes on only one value at a time.
Example 1. c = 0. Solve this quadratic equation: ax² + bx = 0 Solution. Since there is no constant term -- c = 0 -- x is a common factor:
Those are the two roots. Problem 3. Find the roots of each quadratic.
Example 2. b = 0. Solve this quadratic equation: ax² − c = 0. Solution. In the case where there is no middle term, we can write:
However, if the form is the difference of two squares -- x² − 16 -- then we can factor: (x + 4)(x −4) The roots are ±4. In fact, if the quadratic is x² − c, then we could factor: (x + )(x − ) so that the roots are ±. Problem 4. Find the roots of each quadratic.
Example 3. Solve this quadratic equation:
Thus, an equation is solved when x is isolated on the left. Problem 5. Solve each equation for x.
Example 4. Solve this equation
Solution. We can put this equation in the standard form by changing all the signs on both sides. 0 will not change. We have the standard form:
Next, we can get rid of the fraction by multiplying both sides by 2. Again, 0 will not change.
Problem 6. Solve for x.
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