1.5: Quadratic Equations
Warm Up:
Write the following expression in standard form:
Logistical stuff:
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-Math XL in IC
-People with documentation for extended time
2−3𝑖
4+𝑖
Quadratic Equations
Examples:
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2
5𝑥 + 3𝑥 − 1 = 0
−2𝑥 2 − 6𝑥 + 7 = 0
4𝑥 2 + 1 = 0
𝑥2 + 𝑥 = 0
Let’s assume we’re working in the
complex set of numbers this section.
As you can see, a quadratic equation only has one variable (often x),
and is therefore solvable. We will look at 4 different solving techniques:
factoring, square rooting, completing the square, and the quadratic
formula.
Factoring
1. Solve the following quadratic equations by factoring.
a. 𝑥 2 + 11𝑥 = 26
b. 6𝑤 2 + 8𝑤 − 8 = 0
c. 20𝑡 2 − 27𝑡 + 9 = 0
Square Root Property
In certain special cases, we can apply the very simple square root
property to a quadratic equation.
Examples of when we can do this:
 4𝑥 2 − 16 = 0
 6𝑟 2 = 40
 (𝑥 + 1)2 = 9
2. Solve the following quadratic equations by applying the square
root property.
a. 5𝑥 2 − 75 = 0
b. 6ℎ2 + 18 = 3
c. (2𝑧 + 3)2 = 8
3. Solve the following quadratic equation using your graphing
calculator:
0 = 3𝑥 2 − 3𝑥 − 14
We can only apply the square root property when the quadratic
equation is in a special form. Wouldn’t it be nice if we knew some sort
of trick to put ANY quadratic equation into that special form? Do you
think I’d be asking this question if the trick didn’t exist?
Completing the Square
4. Solve the following quadratic equations by completing the square
a. 𝑥 2 + 8𝑥 + 15 = 0
b. 𝑥 2 − 5𝑥 − 14 = 0
c. 𝑚2 + 8𝑚 = 10
d. 𝑡 2 − 7𝑡 = 9
e. 𝑥 2 − 6𝑥 + 17 = 0
f. 2𝑥 2 + 6𝑥 + 3 = 0
g. 3𝑦 2 − 8𝑦 = 18
h. 3𝑣 2 − 5𝑣 − 10 = 0
Let’s apply the completing the square method to our general quadratic
equation, 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0, and see what happens:
Quadratic Formula
There’s even a song!
The Discrimi-WHA?!
5. Solve the following quadratic equations using the quadratic
formula
a. 𝑥 2 − 3𝑥 − 10 = 0
b. 5𝑥 2 + 10𝑥 + 5 = 0
c. 3𝑑2 = 6𝑑 − 1
d. 2𝑧 2 + 5𝑧 + 11 = 0
Let’s summarize. When is it appropriate to use each method (other
than when the teacher tells you specifically)?