Answer :
Let's solve the equation step by step:
The given equation is:
[tex]\[ x^4 + 3 = 13 - 9x^4 \][/tex]
1. Simplify and Rearrange:
- First, move all the terms involving [tex]\(x^4\)[/tex] to one side of the equation:
[tex]\[ x^4 + 9x^4 + 3 = 13 \][/tex]
- Combine the like terms:
[tex]\[ 10x^4 + 3 = 13 \][/tex]
2. Solve for [tex]\(x^4\)[/tex]:
- Subtract 3 from both sides to isolate the term with [tex]\(x^4\)[/tex]:
[tex]\[ 10x^4 = 10 \][/tex]
- Divide both sides by 10 to solve for [tex]\(x^4\)[/tex]:
[tex]\[ x^4 = 1 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Find the fourth roots of 1. The fourth roots of 1 are the numbers that satisfy [tex]\(x^4 = 1\)[/tex]. These are:
[tex]\[ x = 1, -1, i, -i \][/tex]
- Here, [tex]\(i\)[/tex] is the imaginary unit, where [tex]\(i^2 = -1\)[/tex].
Therefore, the solutions to the equation are [tex]\(x = 1, -1, i, -i\)[/tex].
The given equation is:
[tex]\[ x^4 + 3 = 13 - 9x^4 \][/tex]
1. Simplify and Rearrange:
- First, move all the terms involving [tex]\(x^4\)[/tex] to one side of the equation:
[tex]\[ x^4 + 9x^4 + 3 = 13 \][/tex]
- Combine the like terms:
[tex]\[ 10x^4 + 3 = 13 \][/tex]
2. Solve for [tex]\(x^4\)[/tex]:
- Subtract 3 from both sides to isolate the term with [tex]\(x^4\)[/tex]:
[tex]\[ 10x^4 = 10 \][/tex]
- Divide both sides by 10 to solve for [tex]\(x^4\)[/tex]:
[tex]\[ x^4 = 1 \][/tex]
3. Solve for [tex]\(x\)[/tex]:
- Find the fourth roots of 1. The fourth roots of 1 are the numbers that satisfy [tex]\(x^4 = 1\)[/tex]. These are:
[tex]\[ x = 1, -1, i, -i \][/tex]
- Here, [tex]\(i\)[/tex] is the imaginary unit, where [tex]\(i^2 = -1\)[/tex].
Therefore, the solutions to the equation are [tex]\(x = 1, -1, i, -i\)[/tex].
