Question
Solve the following quadratic equation using square roots. $\frac {-9+(x-3)^{4}}{4}=-4$ c
$\{ 2,-8\} $
$\{ 28,-22\} $
$\{ 8,-2\} $
$\{ 16,-10\} $
$\{ 2,-8\} $
$\{ 28,-22\} $
$\{ 8,-2\} $
$\{ 16,-10\} $
Answers
Answered by
GPT-4o mini
To solve the equation
\[
\frac{-9 + (x-3)^4}{4} = -4,
\]
we start by eliminating the fraction. We can multiply both sides by 4:
\[
-9 + (x-3)^4 = -16.
\]
Next, we add 9 to both sides:
\[
(x-3)^4 = -16 + 9,
\]
which simplifies to
\[
(x-3)^4 = -7.
\]
Since the left-hand side \((x-3)^4\) represents a fourth power, it is always non-negative (greater than or equal to 0). However, the right-hand side is negative (\(-7\)).
Because a non-negative number cannot equal a negative number, there are no real solutions to the equation
\[
(x-3)^4 = -7.
\]
Thus, we conclude that the solution set is empty, and the quadratic equation has no solutions in the real number system.
However, if this was an inclusion of the options, you should check if the solution might be complex or if typical problems overlook details. Given the setups, no values in the options can satisfy the equation since I determined it has no real solutions. Thus, it appears the problem might be inadvertently set up without real solutions unless clarified otherwise.
\[
\frac{-9 + (x-3)^4}{4} = -4,
\]
we start by eliminating the fraction. We can multiply both sides by 4:
\[
-9 + (x-3)^4 = -16.
\]
Next, we add 9 to both sides:
\[
(x-3)^4 = -16 + 9,
\]
which simplifies to
\[
(x-3)^4 = -7.
\]
Since the left-hand side \((x-3)^4\) represents a fourth power, it is always non-negative (greater than or equal to 0). However, the right-hand side is negative (\(-7\)).
Because a non-negative number cannot equal a negative number, there are no real solutions to the equation
\[
(x-3)^4 = -7.
\]
Thus, we conclude that the solution set is empty, and the quadratic equation has no solutions in the real number system.
However, if this was an inclusion of the options, you should check if the solution might be complex or if typical problems overlook details. Given the setups, no values in the options can satisfy the equation since I determined it has no real solutions. Thus, it appears the problem might be inadvertently set up without real solutions unless clarified otherwise.
Answer
Just pick one of the answers shown
Answered by
GPT-4o mini
Since the equation we found, \((x-3)^4 = -7\), has no real solutions, we should choose an option typically associated with an answer set presented. Since the choices are to select any one of them, we can arbitrarily choose one option.
Let's select:
\[
\{ 2, -8 \}
\]
This choice is arbitrary, as none of the options will satisfy the original equation.
Let's select:
\[
\{ 2, -8 \}
\]
This choice is arbitrary, as none of the options will satisfy the original equation.