Solving Quadratic Equations Worksheet By Using The Quadratic Formula With Answers Page 8
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10leading coefficient has to be 1 and the x - and x-
SOLUTION:
term must be isolated. It is also easier if the
If the graph is tangent to the x-axis, meaning there is
coefficient of the x-term is even; if not, the
only one x-intercept, then there is only one real
calculations become harder when dealing with
9-5 Solving Quadratic Equations by Using the Quadratic Formula
solution.
2
fractions. For example x
+ 4x = 7 can be solved by
completing the square.
56. Both a and b are greater than 0 and c is less than 0
in a quadratic equation.
SOLUTION:
2
The discrimininant is b
– 4ac. No matter the value
2
of b, b
will always be positive. If a is greater than 0
and c is less than 0, then – 4ac will be positive. Thus
Quadratic Formula:
the discrimininant would be positive. So there would
The Quadratic Formula will work for any quadratic
be two real solutions.
equation and exact solutions can be found. This
method can be time consuming, especially if an
58. WRITING IN MATH Describe the advantages
equation is easily factored. For example, use the
and disadvantages of each method of solving
2
Quadratic Formula to find the solutions of f (x) = 4x
quadratic equations. Which method do you prefer,
+ 13 x + 5.
and why?
SOLUTION:
Factoring:
Factoring is easy if the polynomial is factorable and
complicated if it is not. Not all equations are
factorable.
2
2
For example f (x) = x
– 8x + 16 factors to (x – 4)
.
2
However, f (x) = x
– 16x + 8 can not be factored.
See students’ preferences.
Graphing:
Graphing only gives approximate answers, but it is
60. SHORT RESPONSE The triangle shown is an
easy to see the number of solutions. Using square
isosceles triangle. What is the value of x?
roots is easy when there is no x-term.
2
– 17x + 4,
For example, for the quadratic f (x) = 2x
you can see the two solutions in the graph. However,
it will be difficult to identify the solution x =
8.2578049 in the graph.
SOLUTION:
Because an isosceles triangle has two equal angles, x
could be equal to 64, or it could be equal to the
unnamed angle, where 180 = 2x + 64.
[-5, 15] scl: 2 by [-30, 10] scl: 4
The value of x is 58 or 64.
Completing the square:
Completing the square can be used for any quadratic
2
62. What are the solutions of the quadratic equation 6h
equation and exact solutions can be found, but the
+ 6h = 72?
2
leading coefficient has to be 1 and the x
- and x-
A 3 or −4
term must be isolated. It is also easier if the
B −3 or 4
coefficient of the x-term is even; if not, the
C no solution
calculations become harder when dealing with
D 12 or −48
2
fractions. For example x
+ 4x = 7 can be solved by
SOLUTION:
completing the square.
Write the equation in standard form.
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Page 8
For this equation, a = 6, b = 6, and c = –72.
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