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37
QUADRATIC EQUATIONS
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).
Now, if x = −4, then the first factor will be 0. (Lesson 2.) While if x = 2, the second factor will be 0. But if any factor is 0, then the entire product will be 0. Therefore, if x = −4 or 2, then
x² + 2x − 8 = 0.
−4 and 2 are the solutions to the quadratic equation. They are the roots of that quadratic.
Conversely, if the roots are a and b say, 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. (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 (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.
If (x + q) is a factor, then x = −q is a root.
−q + q = 0.
Problem 1. If a quadratic can be factored as (x + 3)(x − 1), then what are the two roots?
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−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 called a double root.
When will a quadratic have a double root? When the quadratic is a perfect square trinomial.
Example 1. Solve for x: 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
| 2x − 1 | = | 0, |
| we must solve that little equation. (Lesson 9.) | ||
| We have: | ||
| 2x | = | 1 |
| x | = | 1 2 |
The solutions are:
| x | = | 1 2 |
or −5. |
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.
| a) | x² − 3x + 2 | b) | x² + 7x + 12 | |
| (x − 1)(x − 2) | (x + 3)(x + 4) | |||
| x = 1 or 2. | x = −3 or −4. | |||
Again, we use the conjunction "or," because x takes on only one value at a time.
| c) | x² + 3x − 10 | d) | x² − x − 30 | |
| (x + 5)(x − 2) | (x + 5)(x − 6) | |||
| x = −5 or 2. | x = −5 or 6. | |||
| e) | 2x² + 7x + 3 | f) | 3x² + x − 2 | |||||
| (2x + 1)(x + 3) | (3x − 2)(x + 1) | |||||||
| x = − | 1 2 |
or −3. | x = | 2 3 |
or −1. | |||
| g) | x² + 12x + 36 | h) | x² − 2x + 1 | |
| (x + 6)² | (x − 1)² | |||
| x = −6, −6. | x = 1, 1. | |||
| A double root. | A double root. | |||
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:
| x(ax + b) | = | 0. | ||||
| This implies: | ||||||
| x | = | 0 | ||||
| or | ||||||
| x | = | − | b a |
. | ||
Those are the two roots.
Problem 4. Find the roots of each quadratic.
| a) | x² − 5x | b) | x² + x | |
| x(x − 5) | x(x + 1) | |||
| x = 0 or 5. | x = 0 or −1. | |||
| c) | 3x² + 4x | d) | 2x² − x | |||
| x(3x + 4) | x(2x − 1) | |||||
| x = 0 or − | 4 3 |
x = 0 or ½ | ||||
Example 3. b = 0. Solve this quadratic equation:
ax² − c = 0.
Solution. In the case where there is no middle term, we can write:
| ax² | = | c. | |
| This implies: | |||
| x² | = | c a |
|
| x | = |  , according to Lesson 26. |
|
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.
| a) | x² − 3 | b) | x² − 25 | c) | x² − 10 | ||
| x² = 3 | (x + 5)(x − 5) | (x +  )(x − ) |
|||||
x = ± . |
x = ±5. | x = ±. |
|||||
Example 4. Solve this quadratic equation:
| x² | = | x + 20. |
| Solution. First, rewrite the equation in the standard form, by transposing all the terms to the left: | ||
| x² − x − 20 | = | 0 |
| (x + 4)(x − 5) | = | 0 |
| x | = | −4 or 5. |
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.
| a) | x² = 5x − 6 | b) | x² + 12 = 8x | |
| x² − 5x + 6 = 0 | x² − 8x + 12 = 0 | |||
| (x − 2)(x − 3) = 0 | (x − 2)(x − 6) = 0 | |||
| x = 2 or 3. | x = 2 or 6. | |||
| c) | 3x² + x = 10 | d) | 2x² = x | |
| 3x² + x − 10 = 0 | 2x² − x = 0 | |||
| (3x − 5)(x + 2) = 0 | x(2x − 1) = 0 | |||
| x = 5/3 or − 2. | x = 0 or 1/2. | |||
Example 5. Solve this equation
| 3 − | 5 2 |
x − 3x² | = | 0 |
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:
| 3x² + | 5 2 |
x − 3 | = | 0 |
Next, we can get rid of the fraction by multiplying both sides by 2. Again, 0 will not change.
| 6x² + 5x − 6 | = | 0 |
| (3x − 2)(2x + 3) | = | 0. |
| The roots are | 2 3 |
and − | 3 2 |
. |
Problem 7. Solve for x.
| a) | 3 − | 11 2 |
x − 5x² | = 0 | b) | 4 + | 11 3 |
x − 5x² | = 0 |
| 5x² + | 11 2 |
x − 3 | = 0 | 5x² − | 11 3 |
x − 4 | = 0 | |
| 10x² + 11x − 6 | = 0 | 15x² − 11x − 12 | = 0 | |||||
| (5x − 2 )(2x + 3) | = 0 | (3x − 4)(5x + 3 ) | = 0 | |||||
| The roots are | 2 5 |
and − | 3 2 |
. | The roots are | 4 3 |
and − | 3 5 |
. |
| c) | −x² − x + 20 | = 0 | d) | −x² + 3x + 18 | = 0 | |
| x² + x − 20 | = 0 | x² − 3x − 18 | = 0 | |||
| (x + 5)(x − 4) | = 0. | (x − 6)(x + 3) | = 0. | |||
| x = −5 or 4. | x = 6 or −3. | |||||
Section 2: Completing the square
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