IDNLearn.com: Your trusted source for finding accurate answers. Ask your questions and get detailed, reliable answers from our community of knowledgeable experts.
Sagot :
To solve the equation [tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex], we need to follow these steps:
1. Expand both sides:
[tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex]
2. First, expand the left-hand side:
[tex]\((x + 1)(x^2 + x + 1)\)[/tex] and [tex]\((x - 1)(x^2 + x + 1)\)[/tex]
Let's expand [tex]\((x + 1)(x^2 + x + 1)\)[/tex]:
[tex]\[ (x + 1)(x^2 + x + 1) = x(x^2 + x + 1) + 1(x^2 + x + 1) = x^3 + x^2 + x + x^2 + x + 1 = x^3 + 2x^2 + 2x + 1 \][/tex]
Now expand [tex]\((x - 1)(x^2 + x + 1)\)[/tex]:
[tex]\[ (x - 1)(x^2 + x + 1) = x(x^2 + x + 1) - 1(x^2 + x + 1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1 \][/tex]
3. Combine the expanded forms:
Now substitute these back into the equation:
[tex]\[ x^3 + 2x^2 + 2x + 1 - (x^3 - 1) = 2(x - 2) \][/tex]
Simplify the left-hand side:
[tex]\[ (x^3 + 2x^2 + 2x + 1) - x^3 + 1 = 2(x - 2) \][/tex]
Combine like terms on the left:
[tex]\[ x^3 + 2x^2 + 2x + 1 - x^3 + 1 = 2(x - 2) \][/tex]
[tex]\[ 2x^2 + 2x + 2 = 2(x - 2) \][/tex]
4. Simplify further:
We notice that [tex]\(2\)[/tex] is a common factor on the left side:
[tex]\[ 2(x^2 + x + 1) = 2(x - 2) \][/tex]
Divide both sides by 2:
[tex]\[ x^2 + x + 1 = x - 2 \][/tex]
5. Move all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 + x + 1 - x + 2 = 0 \][/tex]
Simplify:
[tex]\[ x^2 + 3 = 0 \][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[ x^2 + 3 = 0 \][/tex]
Subtract 3 from both sides:
[tex]\[ x^2 = -3 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \sqrt{-3} \][/tex]
Since [tex]\(\sqrt{-3} = \sqrt{3}i\)[/tex], we have:
[tex]\[ x = \pm \sqrt{3}i \][/tex]
So, the solution to the equation [tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex] is:
[tex]\[ x = \pm \sqrt{3}i \][/tex]
Or more explicitly:
[tex]\[ x = -\sqrt{3}i \quad \text{and} \quad x = \sqrt{3}i \][/tex]
Thus, the solutions are [tex]\(x = -\sqrt{3}i\)[/tex] and [tex]\(x = \sqrt{3}i\)[/tex].
1. Expand both sides:
[tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex]
2. First, expand the left-hand side:
[tex]\((x + 1)(x^2 + x + 1)\)[/tex] and [tex]\((x - 1)(x^2 + x + 1)\)[/tex]
Let's expand [tex]\((x + 1)(x^2 + x + 1)\)[/tex]:
[tex]\[ (x + 1)(x^2 + x + 1) = x(x^2 + x + 1) + 1(x^2 + x + 1) = x^3 + x^2 + x + x^2 + x + 1 = x^3 + 2x^2 + 2x + 1 \][/tex]
Now expand [tex]\((x - 1)(x^2 + x + 1)\)[/tex]:
[tex]\[ (x - 1)(x^2 + x + 1) = x(x^2 + x + 1) - 1(x^2 + x + 1) = x^3 + x^2 + x - x^2 - x - 1 = x^3 - 1 \][/tex]
3. Combine the expanded forms:
Now substitute these back into the equation:
[tex]\[ x^3 + 2x^2 + 2x + 1 - (x^3 - 1) = 2(x - 2) \][/tex]
Simplify the left-hand side:
[tex]\[ (x^3 + 2x^2 + 2x + 1) - x^3 + 1 = 2(x - 2) \][/tex]
Combine like terms on the left:
[tex]\[ x^3 + 2x^2 + 2x + 1 - x^3 + 1 = 2(x - 2) \][/tex]
[tex]\[ 2x^2 + 2x + 2 = 2(x - 2) \][/tex]
4. Simplify further:
We notice that [tex]\(2\)[/tex] is a common factor on the left side:
[tex]\[ 2(x^2 + x + 1) = 2(x - 2) \][/tex]
Divide both sides by 2:
[tex]\[ x^2 + x + 1 = x - 2 \][/tex]
5. Move all terms to one side to form a standard quadratic equation:
[tex]\[ x^2 + x + 1 - x + 2 = 0 \][/tex]
Simplify:
[tex]\[ x^2 + 3 = 0 \][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[ x^2 + 3 = 0 \][/tex]
Subtract 3 from both sides:
[tex]\[ x^2 = -3 \][/tex]
Take the square root of both sides:
[tex]\[ x = \pm \sqrt{-3} \][/tex]
Since [tex]\(\sqrt{-3} = \sqrt{3}i\)[/tex], we have:
[tex]\[ x = \pm \sqrt{3}i \][/tex]
So, the solution to the equation [tex]\((x + 1)(x^2 + x + 1) - (x - 1)(x^2 + x + 1) = 2(x - 2)\)[/tex] is:
[tex]\[ x = \pm \sqrt{3}i \][/tex]
Or more explicitly:
[tex]\[ x = -\sqrt{3}i \quad \text{and} \quad x = \sqrt{3}i \][/tex]
Thus, the solutions are [tex]\(x = -\sqrt{3}i\)[/tex] and [tex]\(x = \sqrt{3}i\)[/tex].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.