To find an equivalent equation, we will look for the roots of the given polynomial equation:
[tex]\[ f(x) = 16x^4 - 81 = 0 \][/tex]
### Step-by-Step Solution
#### Step 1: Recognize the Form
The equation [tex]\(16x^4 - 81 = 0\)[/tex] is a difference of squares. Recall that a difference of squares follows the pattern:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
#### Step 2: Express as a Difference of Squares
We can express [tex]\(16x^4 - 81\)[/tex] as a difference of squares. Notice that:
[tex]\[ 16x^4 = (4x^2)^2 \][/tex]
[tex]\[ 81 = 9^2 \][/tex]
So, we can rewrite the equation as:
[tex]\[ (4x^2)^2 - 9^2 = 0 \][/tex]
#### Step 3: Apply the Difference of Squares Formula
Now apply the difference of squares formula:
[tex]\[ (4x^2 - 9)(4x^2 + 9) = 0 \][/tex]
#### Step 4: Solve Each Factor
To find the roots, solve each factor separately.
1. Solve [tex]\(4x^2 - 9 = 0\)[/tex]:
[tex]\[
4x^2 - 9 = 0
\][/tex]
[tex]\[
4x^2 = 9
\][/tex]
[tex]\[
x^2 = \frac{9}{4}
\][/tex]
[tex]\[
x = \pm \frac{3}{2}
\][/tex]
2. Solve [tex]\(4x^2 + 9 = 0\)[/tex]:
[tex]\[
4x^2 + 9 = 0
\][/tex]
[tex]\[
4x^2 = -9
\][/tex]
[tex]\[
x^2 = -\frac{9}{4}
\][/tex]
[tex]\[
x = \pm \frac{3i}{2}
\][/tex]
#### Step 5: Combine All Roots
The roots of the equation [tex]\(16x^4 - 81 = 0\)[/tex] are:
[tex]\[ x = -\frac{3}{2}, \frac{3}{2}, -\frac{3i}{2}, \frac{3i}{2} \][/tex]
Therefore, an equivalent set of equations would be:
[tex]\[ x = -\frac{3}{2} \][/tex]
[tex]\[ x = \frac{3}{2} \][/tex]
[tex]\[ x = -\frac{3i}{2} \][/tex]
[tex]\[ x = \frac{3i}{2} \][/tex]
This means the equivalent equation capturing all these roots is exactly the given polynomial equation:
[tex]\[ 16x^4 - 81 = 0 \][/tex]
