Download Transcription -- Part I

Quadratic Equations
Section 1.3
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Quadratic Equation – Can be written in the form ax2 + bx + c = 0 where a, b, and c are real
numbers and a cannot equal 0.
Zero Factor Theorem – if a and b are real numbers and if ab = 0, then a=0 or b=0.
Ex: Solve 2x2 + 9x -35 = 0
First, let’s factor it. Factoring gives:
(2x + 5)(x – 7) = 0
So either:
2x + 5 = 0 or x – 7 = 0
Now we solve each of these equations. Subtract 5 from both sides on the left and add 7 to both
sides on the right:
2x = -5 or x = 7
We’re not quite done with the left one. We divide by two on both sides.
x = -5/2
Our answers are x = -5/2 or x = 7.
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Square Root Property – If c is greater than 0, the equation x2 = c has two real roots: x = the
square root of c, or x = the negative square root of c.
Ex: Solve x2 – 7 = 0.
We want it to look like x2 = c, so we’re going to add 7 to both sides.
x2 = 7
Now we’re going to use the square root property, so that means:
x = square root 7 or x = negative square root 7
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Completing the Square:
1. If the coefficient of x2 is not 1, change it to 1 by dividing both sides by the current
coefficient of x2.
2. Move the constant to the right hand side.
3. Complete the square on x:
a. Take one-half of the coefficient of x, and then square it.
b. Add this number to both sides of the equation.
4. Factor the perfect square trinomial (it will always be (x + the number in 3a)***keep the
sign from 3a***) and combine like terms.
5. Solve the resulting quadratic equation by square root property.
Ex: Solve by completing the square: x2 – 2x – 9 = 0.
The first thing we check, is the coefficient of x2 1? It is, so we move to number 2. Move
the constant to the right-hand side.
x2 – 2x – 9 + 9 = 0 + 9
x2 – 2x = 9
We’re going to complete the square on x. We look at the coefficient on x. It’s -2, so
we’re going to multiply it by ½ and square it.
(-2)(1/2) = -1
Squaring:
(-1)2 = 1
Add this to both sides. I’m going to leave this as (-1)2 to show how this next part can be
a little bit easier.
x2 -2x + (-1)2 = 9 + (-1)2
Here’s the fun part. This left-hand side is going to factor as a perfect square trinomial.
We did that on purpose. That’s why we took half and then squared it. To factor, we’re
just going to write x and then read of what’s in parentheses, quantity squared. So we
get:
(x – 1)2 = 10
We solve this using the square root property.
x -1 = positive square root 10 or x – 1 = negative square root 10.
Add 1 to both sides:
x = 1 + square root 10, or x = 1 – square root 10
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The Quadratic Formula – The solutions of the general quadratic equation ax2 + bx + c = 0 (a does
not equal 0) are:
x = -b +/- square root (b2 -4ac)
2a
Ex: Use the quadratic formula to solve 4x2 + 16x + 13 = 0.
First, we need to identify what a, b, and c are. a is the coefficient of x2, b is the coefficient of x,
and c is the constant.
a = 4, b = 16, and c = 16
Now we plug it into the formula.
x = -16 +/- square root [(16)2 – 4(4)(13)]
2(4)
x = -16 +/- square root [256 – 208]
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x = -16 +/- square root (48)
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48 is 16 * 3, so we can pull out the square root of 16, which is 4.
x = -16 +/- 4 *square root (3)
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We can write this as:
x = -16 + 4 square root 3 or x = -16 – 4 square root 3
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Remember, a + b all over c is the same as a/c + b/c. Likewise with a – b all over c.
x = -16 + 4 square root 3 or x = -16 – 4 square root 3
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Then we can reduce.
x = -2 + (square root 3)/2 or x = -2 – (square root 3)/2
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Note: You are not responsible for the information involving the Discriminant.
Make sure you can solve quadratic equations using quadratic formula, completing the square,
square root property, or the zero factor theorem.