Answer :
To solve the equation [tex]\(4x^2 + 27 = x^2\)[/tex] using square roots, we can break it down into a step-by-step process:
1. Rearrange the Equation:
Start by moving all terms to one side of the equation to set it to zero:
[tex]\[
4x^2 + 27 - x^2 = 0
\][/tex]
2. Simplify the Equation:
Combine like terms:
[tex]\[
3x^2 + 27 = 0
\][/tex]
3. Isolate the Quadratic Term:
Subtract 27 from both sides to focus on the quadratic term:
[tex]\[
3x^2 = -27
\][/tex]
4. Solve for [tex]\(x^2\)[/tex]:
Divide both sides by 3 to isolate [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = -9
\][/tex]
5. Take the Square Root:
To find [tex]\(x\)[/tex], take the square root of both sides. Remember that taking the square root of a negative number introduces the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
x = \pm \sqrt{-9}
\][/tex]
Since [tex]\(\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = 3i\)[/tex], we have:
[tex]\[
x = \pm 3i
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 3i\)[/tex] and [tex]\(x = -3i\)[/tex]. These solutions are complex numbers because the original equation involved a negative value under the square root.
1. Rearrange the Equation:
Start by moving all terms to one side of the equation to set it to zero:
[tex]\[
4x^2 + 27 - x^2 = 0
\][/tex]
2. Simplify the Equation:
Combine like terms:
[tex]\[
3x^2 + 27 = 0
\][/tex]
3. Isolate the Quadratic Term:
Subtract 27 from both sides to focus on the quadratic term:
[tex]\[
3x^2 = -27
\][/tex]
4. Solve for [tex]\(x^2\)[/tex]:
Divide both sides by 3 to isolate [tex]\(x^2\)[/tex]:
[tex]\[
x^2 = -9
\][/tex]
5. Take the Square Root:
To find [tex]\(x\)[/tex], take the square root of both sides. Remember that taking the square root of a negative number introduces the imaginary unit [tex]\(i\)[/tex]:
[tex]\[
x = \pm \sqrt{-9}
\][/tex]
Since [tex]\(\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = 3i\)[/tex], we have:
[tex]\[
x = \pm 3i
\][/tex]
Thus, the solutions to the equation are [tex]\(x = 3i\)[/tex] and [tex]\(x = -3i\)[/tex]. These solutions are complex numbers because the original equation involved a negative value under the square root.
