S k i l l
The standard form of a quadratic equation
Proof of the quadratic formula
The graph of y = A quadratic: A parabola
A QUADRATIC is a polynomial whose highest exponent is 2.
ax² + bx + c.
The coefficient of x² is called the leading coeffieient.
Question 1. What is the standard form of a quadratic equation?
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, the roots of this quadratic --
x² + 2x − 8
-- are the solutions to
x² + 2x − 8 = 0.
To find the roots, we can factor that quadratic as
(x + 4)(x − 2).
If either factor is 0, then the product will be 0. The first factor will be 0 if x = −4. (Lesson 2.) The second factor will be 0 if x = 2. And so if x = −4 or 2, then
x² + 2x − 8 = 0.
−4, 2 are the roots of that quadratic.
Conversely, if the roots are a or b, then the quadratic can be factored as
(x − a)(x − b).
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.
Question 3. How many roots has a quadratic?
Always two. Because a quadratic (with leading coefficient 1, at least) can always be factored as (x − a)(x − b), and a, b are the two roots.
In other words, when the leading coefficient is 1, the root has the opposite sign of the number in the factor.
−q + q = 0.
Problem 1. If a quadratic can be factored as (x + 3)(x − 1), then what are the two roots?
To see the answer, pass your mouse over the colored area.
−3 or 1.
We say "or," because x can take only one value at a time.
Question 4. What do we mean by a double root?
The two roots are equal. The factors will be (x − a)(x − a), so that the two roots are a, a.
For example, this quadratic
x² − 12x + 36
can be factored as
(x − 6)(x − 6).
If x = 6, then each factor will be 0, and therefore the quadratic will be 0. 6 is a double root.
When will a quadratic have a double root? When the quadratic is a perfect square trinomial.
Example 1. Find the roots of 2x² + 9x − 5.
Solution. That quadratic is factored as follows:
2x² + 9x − 5 = (2x − 1)(x + 5).
Now, it is easy to see that the second factor will be 0 when x = −5.
As for the value of x that will make
The roots are:
Those are the two values of x that will make the quadratic equal to 0.
Problem 2. How is it possible that the product of two factors ab = 0?
Either a = 0 or b = 0.
Solution by factoring
Problem 3. Find the roots of each quadratic by factoring.
Again, we use the conjunction "or," because x takes on only one value at a time.
Example 2. 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 4. Find the roots of each quadratic.
Example 3. 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 it as:
(x + 4)(x −4).
The roots are ±4.
In fact, if the quadratic is
x² − c,
then we could factor it as:
(x + )(x − ),
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 x is isolated on the left.
x = ± is not a solution.
Problem 6. Solve each equation for x.
Example 5. 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 7. Solve for x.
Section 2: Completing the square
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