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In Lessons 12, 13, and 14 we have seen that quadratic equations are
equations that can be written in the form ๐๐ฅ
2
- ๐๐ฅ + ๐ = 0. In this lesson
we will see that certain polynomial equations which are not quadratic can
still be solved using some of the same methods as quadratics. For
instance, the equation ๐ฅ
4
2
โ 8 = 0 is not quadratic because the
degree of the polynomial is 4 rather than 2 ; however, it can still be solved
by factoring, just like a quadratic equation.
Steps for Solving an Equation by Factoring:
- write the equation as a polynomial and set it equal to zero
- factor the polynomial
- use Zero Factor Theorem to solve
Example 1 : Solve the following equations for ๐ฅ and enter exact answers
only (no decimal approximations). If there is more than one solution,
separate your answers with commas. If there are no real solutions, enter
NO SOLUTION.
4
2
4
2
2
2
2
2
2
2
2
2
2
2
2
cannot be equal to โ 2 using real numbers
2
Remember that a
perfect square (such
as ๐ฅ
2
) cannot be
equal to a negative
value when you are
working with real
numbers. So when
you have ๐ฅ
2
= โ 2 ,
you should recognize
that this is not
possible with real
numbers and thus we
eliminate that
equation. If you go
ahead and take the
square root of both
sides of the equation,
you would end up
with = ยฑโโ 2.
Once again this is
not possible with real
numbers, so we
eliminate that
equation.
โ 1 , 8 7
1 , โ 8 โ 7
โ 2 , 4 2
๐, โ๐ โ๐
Always be sure to check that the value that is equal to a perfect
square, or under a square root, is non-negative. Also, keep in mind
that when taking the square root of both sides of an equation, we have
a positive root and a negative root.
Example 2 : Solve the following equations for ๐ฅ and enter exact answers
only (no decimal approximations). If there is more than one solution,
separate your answers with commas. If there are no real solutions, enter
NO SOLUTION.
Any radicals and/or fractions in your final answer should be simplified
completely.
a. ๐ฅ
4
2
โ 54 = 0 b. 75 ๐ฅ
4
3
2
b. ๐
4
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
4
3
2
7
2
2
cannot be equal to โ
4
3
or โ
7
2
using real numbers
1
2
10
3
Example 3 : Solve the following equations for ๐ฅ and enter exact answers
only (no decimal approximations). If there is more than one solution,
separate your answers with commas. If there are no real solutions, enter
NO SOLUTION.
Any radicals and/or fractions in your final answer should be simplified
completely.
4
2
4
2
4
2
+ _____ = 10 + _____
4
2
2
2
4
2
2
2
2
2
2
2
2
cannot be equal to 1 โ โ 11
because 1 โ โ
11 is negative
2
Remember that a
perfect square (such
as ๐ฅ
2
) cannot be
equal to a negative
value when you are
working with real
numbers. So when
you have
๐ฅ
2
= 1 โ โ 11 , you
should recognize that
1 โ โ
11 is a
negative value. If
you donโt, and had
simply taken the
square root of both
sides of the equation,
you would end up
with = ยฑ
โ
1 โ โ 11.
At this point you
would need to
recognize that
โ
1 โ โ 11 does not
exist because once
again, 1 โ โ 11 is a
negative value.
1 , โ 10 โ 9
โ 1 , 10 9
2 , โ 5 โ 3
โ 2 , 5 3
Once again, ALWAYS be sure to check that the value that is equal to
a perfect square, or under a square root, is non-negative.
Final answers should be left in exact form unless the directions state
otherwise. However, it might be helpful to approximate a final answer
using a calculator when working with radicals to determine whether a
radicand is positive or negative.
Example 4: Solve the following equations for ๐ฅ and enter exact answers
only (no decimal approximations). If there is more than one solution,
separate your answers with commas. If there are no real solutions, enter
NO SOLUTION.
Any radicals and/or fractions in your final answer should be simplified
completely.
a. 4 ๐ฅ
4
2
โ 4
โ 2
b. ๐
4 ๐ฅ
4
4
12 ๐ฅ
2
4
4
4
0
4
4
2
4
2
3
2
2
3
2
2
4
2
9
4
9
4
2
3
2
2
5
4
2
3
2
5
4
2
3
2
5
4
2
3
2
5
4
โ
3
2
5
4
โ โ 0. 38 ; โ
3
2
โ โ
5
4
โ โ 2. 62
Since both are negative, neither is possible.
Answers to Exercises:
1a. ๐ฅ = โ 2 , 2 ; 2 a. ๐ฅ = โ 3 โ 2 , 3 โ 2 ; 2 b. ๐ฅ = โ 1 , 0 ,
2
15
2 c. NO SOLUTIONS ; 2d. ๐ฅ = โโ 2 , โโ
3
5
3
5
3 a. ๐ฅ =
1 + โ 11 ; 4a. NO SOLUTIONS ;
4 b. ๐ฅ =
4 c. ๐ฅ =
1 + โ 5 ; 4d. ๐ฅ = 0 , โ
1
10
โ
41
10
1
10
โ
41
10
