IDNLearn.com: Your trusted source for finding accurate and reliable answers. Join our Q&A platform to get accurate and thorough answers to all your pressing questions.
Sagot :
To solve the equation [tex]\(x^2 + 2x - 9 = 15\)[/tex] by completing the square, follow these steps:
1. Isolate the constant term:
Move all terms to one side of the equation to isolate the constant.
[tex]\[ x^2 + 2x - 9 - 15 = 0 \][/tex]
Simplify this to:
[tex]\[ x^2 + 2x - 24 = 0 \][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[ x^2 + 2x = 24 \][/tex]
3. Complete the square:
To complete the square, you need to add and subtract the same value to/from the left side of the equation. Here, focus on the [tex]\(x^2 + 2x\)[/tex] part.
Take half of the coefficient of [tex]\(x\)[/tex], square it, and add it inside the equation:
- Coefficient of [tex]\(x\)[/tex] is 2
- Half of it: [tex]\(2/2 = 1\)[/tex]
- Squaring it: [tex]\(1^2 = 1\)[/tex]
Add and subtract 1 to/from the left side:
[tex]\[ x^2 + 2x + 1 - 1 = 24 \][/tex]
This simplifies to:
[tex]\[ (x + 1)^2 - 1 = 24 \][/tex]
4. Simplify the equation:
Move the -1 to the right side:
[tex]\[ (x + 1)^2 = 25 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[ x + 1 = \pm 5 \][/tex]
Solve for [tex]\(x\)[/tex] in both cases:
- [tex]\(x + 1 = 5\)[/tex]
[tex]\[ x = 5 - 1 \][/tex]
[tex]\[ x = 4 \][/tex]
- [tex]\(x + 1 = -5\)[/tex]
[tex]\[ x = -5 - 1 \][/tex]
[tex]\[ x = -6 \][/tex]
So, the solutions to the equation are [tex]\(x = 4\)[/tex] and [tex]\(x = -6\)[/tex]. Thus, the correct answer is:
C. [tex]\(x = -6\)[/tex]; [tex]\(x = 4\)[/tex]
1. Isolate the constant term:
Move all terms to one side of the equation to isolate the constant.
[tex]\[ x^2 + 2x - 9 - 15 = 0 \][/tex]
Simplify this to:
[tex]\[ x^2 + 2x - 24 = 0 \][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[ x^2 + 2x = 24 \][/tex]
3. Complete the square:
To complete the square, you need to add and subtract the same value to/from the left side of the equation. Here, focus on the [tex]\(x^2 + 2x\)[/tex] part.
Take half of the coefficient of [tex]\(x\)[/tex], square it, and add it inside the equation:
- Coefficient of [tex]\(x\)[/tex] is 2
- Half of it: [tex]\(2/2 = 1\)[/tex]
- Squaring it: [tex]\(1^2 = 1\)[/tex]
Add and subtract 1 to/from the left side:
[tex]\[ x^2 + 2x + 1 - 1 = 24 \][/tex]
This simplifies to:
[tex]\[ (x + 1)^2 - 1 = 24 \][/tex]
4. Simplify the equation:
Move the -1 to the right side:
[tex]\[ (x + 1)^2 = 25 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Take the square root of both sides:
[tex]\[ x + 1 = \pm 5 \][/tex]
Solve for [tex]\(x\)[/tex] in both cases:
- [tex]\(x + 1 = 5\)[/tex]
[tex]\[ x = 5 - 1 \][/tex]
[tex]\[ x = 4 \][/tex]
- [tex]\(x + 1 = -5\)[/tex]
[tex]\[ x = -5 - 1 \][/tex]
[tex]\[ x = -6 \][/tex]
So, the solutions to the equation are [tex]\(x = 4\)[/tex] and [tex]\(x = -6\)[/tex]. Thus, the correct answer is:
C. [tex]\(x = -6\)[/tex]; [tex]\(x = 4\)[/tex]
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find the answers you need at IDNLearn.com. Thanks for stopping by, and come back soon for more valuable insights.