Download 5-5 Solving Quadratic Equations

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5-5 Solving Quadratic Equations
There are many ways we can solve a quadratic equation. The first is by factoring.
1) Make sure the entire quadratic equation equals to zero.
2) Factor. Then set each factor equal to zero & solve.
(Remember to always look for a GCF first!)
Example 1… Solve each equation by factoring.
a. x2 – 7x – 18 = 0
b. 2x2 – 4x = 6
c. 3x2 – 20x – 7 = 0
d. 3x2 = -5x + 12
e. 3x2 + 12x + 12 = 0
f. x2 – 64 = 0
Sometimes, we can bypass the whole factoring thing and just solve by finding square roots.
Example 2… Solve each quadratic equation by finding square roots.
a. x2 – 25 = 0
b. 4x2 – 49 = 0
c. The tallest building in the world (according to Wikipedia) is the Burj Kalifah in Dubai. It stands
2,722 feet tall. The function
models the height y in feet of an object t
seconds after it is dropped from the top of the building. How long will it take the object to hit
the ground?
5-8 The Quadratic Formula
Example 1… Simplify each square root as much as possible.
a.
b.
c.
Another way of solving quadratic equations is to use the quadratic formula. Unlike factoring,
the quadratic formula ALWAYS works.
The quadratic formula is :
Example 2… Solve each quadratic equation using the quadratic formula. (Remember to
first make sure the whole quadratic equation is equal to zero!)
a. x2 – 4x + 3 = 0
b. x2 = 6x – 1
c. 2x2 + 7x + 5 = 0
d. x2 + 9x – 18 = 0
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5-6 Complex Numbers
An imaginary number is defined as the square root of a negative number.
and
A complex number is a combination of a real number and an imaginary number.
For example,
or
Example 1… Simplify.
a.
b.
c.
d.
e.
f.
Example 2… Find the additive inverse (opposite) of each complex number.
a.
b. –
Example 3… Add or subtract the following complex numbers.
a.
b.
Example 4… Multiply the following complex numbers.
a.
b.
Example 5… Graph each complex number and find the absolute value of each complex
number. (Remember that absolute value means distance from zero!)
a. |-7|
b. |2i|
c. |3 – 4i|
d. |-5+6i|
Example 6… Solve the quadratic equation by finding square roots.
a. x2 = -25
b. 3x2 + 48 = 0
The discriminant, b²-4ac, is used to help us find out how many and what type of solutions the
quadratic function will have. There are three options:
1) If the discriminant is > 0, there will be 2 real solutions.
2) If the discriminant is = 0, there will be 1 real solution.
3) If the discriminant is < 0, there will be 2 imaginary solutions.
Example 7… Use the discriminant to find the number and type of solutions for the
following quadratic equations:
a.
b.
c.
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5-7 Completing the Square
Yet another method of solving quadratic equations is by completing the square.
Example 1… Solve the quadratic equation
.
Completing the square is the process used in order to transform any quadratic equation into
the equation in Example 1, so that we can solve by finding square roots.
Example 2… Complete the square for each expression, to create a perfect square
trinomial. Then factor the expression.
a. x2 + 2x + _____
b. x2 – 12x + _____
c. x2 + 5x + _____
Example 3… Solve each quadratic equation by completing the square.
a. x2 – 12x + 36 = 0
b. x2 + 6x – 12 = 0
c. 2x2 + 12x = -5
d. 3x2 – 9x – 30 = 0