Solving Quadratic Equations Worksheet By Using The Quadratic Formula With Answers Page 7
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10c . You could write an equation with three roots by
multiplying the corresponding factors together and
setting it equal to zero. If an equation has the three
roots 1, 2, 3, then the corresponding factors would be
The solutions are 3.4 and –1.4.
9-5 Solving Quadratic Equations by Using the Quadratic Formula
(x
– 1), (x – 2), and (x – 3). The equation would then
50. MULTIPLE REPRESENTATIONS In this
be:
problem, you will investigate writing a quadratic
equation with given roots.
2
If p is a root of 0 = ax
+ bx + c, then (x – p ) is a
2
factor of ax
+ bx + c.
This is not a quadratic equation since it is of degree
3.
52. REASONING Use factoring techniques to
a. Tabular Copy and complete the first two columns
2
determine the number of real zeros of f (x) = x
− 8x
of the table.
+ 16. Compare this method to using the discriminant.
b. Algebraic Multiply the factors to write each
equation with integral coefficients. Use the equations
SOLUTION:
to complete the last column of the table. Write each
2
For f(x) = x
− 8x + 16, a = 1, b =
−8 and c = 16.
equation.
2
2
Then the discriminate is b
− 4ac or (−8)
−4(1)(16)
c. Analytical How could you write an equation with
three roots? Test your conjecture by writing an
= 0. The polynomial can be factored to get f (x) = (x
equation with roots 1, 2, and 3. Is the equation
2.
− 4)
Solve to find the real zeros.
quadratic? Explain.
SOLUTION:
a . For any two roots m and p , in the left hand column,
the middle column will be the corresponding factors
(x
–
m), (x
–
p ).
So the only real zero is 4. The discriminant is 0, so
b . The equation with these factors will be: (x
–
m)(x
the only real zero is 4. The discriminant is 0, so there
2
–
p ) = 0 which simplifies to x
– (m + p )x + mp = 0.
is 1 real zero. The discriminant tells us how many
Use this to fill in the column of the table.
real zeros there are. Factoring tells us what they are.
CCSS STRUCTURE Determine whether there
are two, one, or no real solutions.
54. The graph of a quadratic function touches but does
not cross the x-axis.
SOLUTION:
If the graph is tangent to the x-axis, meaning there is
only one x-intercept, then there is only one real
solution.
c . You could write an equation with three roots by
multiplying the corresponding factors together and
56. Both a and b are greater than 0 and c is less than 0
setting it equal to zero. If an equation has the three
in a quadratic equation.
roots 1, 2, 3, then the corresponding factors would be
SOLUTION:
(x
– 1), (x – 2), and (x – 3). The equation would then
2
be:
The discrimininant is b
– 4ac. No matter the value
2
will always be positive. If a is greater than 0
of b, b
and c is less than 0, then – 4ac will be positive. Thus
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the discrimininant would be positive. So there would
be two real solutions.
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