Quadratic Functions
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Solving by the Quadratic Formula
2
Example 1 – Using the quadratic formula
Solve the following quadratic equations. Round your
answers to three decimal places. Check your answers.
a. 3x2 – 6x – 24 = 0
b. 4t2 – 8t + 5 = 50
c. 3.4x2 + 4.2x – 7.8 = 0
Solution:
a.
3x2 – 6x – 24 = 0
The equation is in standard form
and equal to zero, so use the
quadratic formula.
3
Example 1 – Solution
cont’d
Separate into two equations.
Simplify.
4
Example 1 – Solution
x=4
cont’d
x = –2
5
Example 1 – Solution
b.
4t2 – 8t + 5 = 50
cont’d
Set the equation equal to zero.
4t2 – 8t – 45 = 0
Use the quadratic formula.
6
Example 1 – Solution
cont’d
Separate into two equations
Simplify
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Example 1 – Solution
c.
cont’d
3.4x2 + 4.2x – 7.8 = 0
Separate into two equations.
Simplify.
8
Example 1 – Solution
x  1.018
cont’d
x  –2.253
Check both answers using the
graph.
Both answers work.
9
Determining Which Algebraic Method to Use
When Solving a Quadratic Equation
10
Determining Which Algebraic Method to Use When Solving a Quadratic Equation
We have looked at four algebraic methods to solve a
quadratic equation:
• Square root property
• Completing the square
• Factoring
• Quadratic formula
Determining what method is best depends on the
characteristics of the equation we are trying to solve.
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Determining Which Algebraic Method to Use When Solving a Quadratic Equation
Square Root Property
The square root property works best when the quadratic is
in vertex form or when there is a squared variable term but
no other variable terms.
(x + 5)2 – 9 = 16
x2 + 13 = 49
6x2 – 4 = 18
In all of these equations, the square root property would be
a good method to use.
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Determining Which Algebraic Method to Use When Solving a Quadratic Equation
Remember, when solving using the square root property, to
isolate the squared variable expression on one side before
using the square root property. Also do not forget to use the
plus/minus symbol to indicate all possible answers.
Completing the square
Completing the square works well for equations that have
both a squared term and a first-degree term. This method is
usually easiest if the numbers are not too large and the
leading coefficient is 1.
13
Determining Which Algebraic Method to Use When Solving a Quadratic Equation
x2 + 4x + 9 = 0
x2 – 5x = 20
After completing the square, we will again use the square
root property to solve.
Factoring
Factoring works best when the numbers are not too large
or when the terms are higher than second degree. Always
remember to first factor out the greatest common factor.
Although factoring does not work with all quadratics, if the
equation factors easily, it can be one of the fastest solution
methods.
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Determining Which Algebraic Method to Use When Solving a Quadratic Equation
For factoring to be used when solving, the equation must
be equal to zero so that the zero product property can be
used.
x2 + 5x + 6 = 0
x3 + x2 – 6x = 0
Equations that cannot be factored may still have solutions,
so use one of the other methods, such as completing the
square or the quadratic formula.
Occasionally, an equation can be factored partially, and
then the separate pieces can be solved using another
method.
15
Determining Which Algebraic Method to Use When Solving a Quadratic Equation
Quadratic Formula
The quadratic formula will work with any quadratic, but it is
easiest if the quadratic starts out in standard form.
Because the quadratic formula basically requires
substituting values for a, b, and c and then simplifying an
arithmetic expression, the formula will work equally well for
large or small numbers.
Whenever a quadratic equation has decimals or fractions,
the quadratic formula is probably the best method to
choose.
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Determining Which Algebraic Method to Use When Solving a Quadratic Equation
5x2 + 16x – 85 = 0
0.25x2 – 3.4x + 9 = 0
In solving with the square root property or the quadratic
formula, a negative under the square root will indicate no
real solutions.
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