Find expert answers and community support for all your questions on IDNLearn.com. Ask your questions and receive detailed and reliable answers from our experienced and knowledgeable community members.
Sagot :
To identify which of the given equations are quadratic and to determine the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for each quadratic equation, we need to follow these steps:
1. Expand and simplify each equation.
2. Check if the resulting equation is in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
3. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] if the equation is quadratic.
Recall that a quadratic equation has the highest degree of 2.
Here are the given equations and the steps taken to determine if they are quadratic equations:
1. [tex]\(x^3 - 6y + 2 = 0\)[/tex]
- This equation has an [tex]\(x^3\)[/tex] term, making it a cubic equation, not quadratic.
2. [tex]\((x - 1)(x + 2) = x(x + 5)\)[/tex]
- Expand both sides:
[tex]\[ (x - 1)(x + 2) \implies x^2 + 2x - x - 2 = x^2 + x - 2 \][/tex]
[tex]\[ x(x + 5) \implies x^2 + 5x \][/tex]
- Setting them equal to each other:
[tex]\[ x^2 + x - 2 = x^2 + 5x \][/tex]
- Subtract [tex]\(x^2 + 5x\)[/tex] from both sides:
[tex]\[ x - 2 = 5x \][/tex]
- Simplify:
[tex]\[ -2 = 4x \quad \text{or} \quad x = -\frac{1}{2} \][/tex]
- This simplifies to a linear equation, not quadratic.
3. [tex]\(20 + 5 = 0\)[/tex]
- This is a constant equation and not a quadratic equation.
4. [tex]\((x + 2)(x - 3) = 5\)[/tex]
- Expand the left side:
[tex]\[ (x + 2)(x - 3) \implies x^2 - 3x + 2x - 6 = x^2 - x - 6 \][/tex]
- Set equal to 5:
[tex]\[ x^2 - x - 6 = 5 \][/tex]
- Rearrange to standard quadratic form:
[tex]\[ x^2 - x - 6 - 5 = 0 \implies x^2 - x - 11 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex].
5. [tex]\(12x^3 + 7x = 15\)[/tex]
- This equation has [tex]\(x^3\)[/tex] term, making it a cubic equation, not quadratic.
6. [tex]\(\left(x^2 + 1 \times 10\right) = 0\)[/tex]
- Simplifies to:
[tex]\[ x^2 + 10 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 10\)[/tex].
7. [tex]\(4(x + 1)^2 = 2(x - 1)\)[/tex]
- Expand and simplify:
[tex]\[ 4(x^2 + 2x + 1) = 4x^2 + 8x + 4 \][/tex]
[tex]\[ 2x - 2 \][/tex]
- Set equal to 0:
[tex]\[ 4x^2 + 8x + 4 - 2x + 2 = 0 \][/tex]
[tex]\[ 4x^2 + 6x + 6 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 4\)[/tex], [tex]\(b = 6\)[/tex], [tex]\(c = 6\)[/tex].
8. [tex]\((x - 1) + 3 = 2x + 1\)[/tex]
- Simplify and rearrange:
[tex]\[ x + 2 = 2x + 1 \][/tex]
[tex]\[ x = 1 \][/tex]
- This is a linear equation, not quadratic.
9. [tex]\(x^2 + 2x + 1 = 5x + 1\)[/tex]
- Set equal to zero:
[tex]\[ x^2 + 2x + 1 - 5x - 1 = 0 \implies x^2 - 3x = 0 \][/tex]
- This simplifies to a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = 0\)[/tex].
10. [tex]\((x + 2)^3 = x(x^2 - 10x + 25)\)[/tex]
- Expand and simplify both sides:
[tex]\[ (x + 2)^3 \implies x^3 + 6x^2 + 12x + 8 \][/tex]
[tex]\[ x(x^2 - 10x + 25) \implies x^3 - 10x^2 + 25x \][/tex]
- Equate and simplify:
[tex]\[ x^3 + 6x^2 + 12x + 8 = x^3 - 10x^2 + 25x \][/tex]
[tex]\[ 16x^2 - 13x + 8 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex].
Summarizing the quadratic equations and their coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\((x + 2)(x - 3) = 5\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
2. [tex]\(\left(x^2 + 1 \times 10\right) = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 10\)[/tex]
3. [tex]\(4(x + 1)^2 = 2(x - 1)\)[/tex]
- [tex]\(a = 4\)[/tex], [tex]\(b = 6\)[/tex], [tex]\(c = 6\)[/tex]
4. [tex]\(x^2 + 2x + 1 = 5x + 1\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = 0\)[/tex]
5. [tex]\((x + 2)^3 = x(x^2 - 10x + 25)\)[/tex]
- [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
1. Expand and simplify each equation.
2. Check if the resulting equation is in the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
3. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] if the equation is quadratic.
Recall that a quadratic equation has the highest degree of 2.
Here are the given equations and the steps taken to determine if they are quadratic equations:
1. [tex]\(x^3 - 6y + 2 = 0\)[/tex]
- This equation has an [tex]\(x^3\)[/tex] term, making it a cubic equation, not quadratic.
2. [tex]\((x - 1)(x + 2) = x(x + 5)\)[/tex]
- Expand both sides:
[tex]\[ (x - 1)(x + 2) \implies x^2 + 2x - x - 2 = x^2 + x - 2 \][/tex]
[tex]\[ x(x + 5) \implies x^2 + 5x \][/tex]
- Setting them equal to each other:
[tex]\[ x^2 + x - 2 = x^2 + 5x \][/tex]
- Subtract [tex]\(x^2 + 5x\)[/tex] from both sides:
[tex]\[ x - 2 = 5x \][/tex]
- Simplify:
[tex]\[ -2 = 4x \quad \text{or} \quad x = -\frac{1}{2} \][/tex]
- This simplifies to a linear equation, not quadratic.
3. [tex]\(20 + 5 = 0\)[/tex]
- This is a constant equation and not a quadratic equation.
4. [tex]\((x + 2)(x - 3) = 5\)[/tex]
- Expand the left side:
[tex]\[ (x + 2)(x - 3) \implies x^2 - 3x + 2x - 6 = x^2 - x - 6 \][/tex]
- Set equal to 5:
[tex]\[ x^2 - x - 6 = 5 \][/tex]
- Rearrange to standard quadratic form:
[tex]\[ x^2 - x - 6 - 5 = 0 \implies x^2 - x - 11 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex].
5. [tex]\(12x^3 + 7x = 15\)[/tex]
- This equation has [tex]\(x^3\)[/tex] term, making it a cubic equation, not quadratic.
6. [tex]\(\left(x^2 + 1 \times 10\right) = 0\)[/tex]
- Simplifies to:
[tex]\[ x^2 + 10 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 10\)[/tex].
7. [tex]\(4(x + 1)^2 = 2(x - 1)\)[/tex]
- Expand and simplify:
[tex]\[ 4(x^2 + 2x + 1) = 4x^2 + 8x + 4 \][/tex]
[tex]\[ 2x - 2 \][/tex]
- Set equal to 0:
[tex]\[ 4x^2 + 8x + 4 - 2x + 2 = 0 \][/tex]
[tex]\[ 4x^2 + 6x + 6 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 4\)[/tex], [tex]\(b = 6\)[/tex], [tex]\(c = 6\)[/tex].
8. [tex]\((x - 1) + 3 = 2x + 1\)[/tex]
- Simplify and rearrange:
[tex]\[ x + 2 = 2x + 1 \][/tex]
[tex]\[ x = 1 \][/tex]
- This is a linear equation, not quadratic.
9. [tex]\(x^2 + 2x + 1 = 5x + 1\)[/tex]
- Set equal to zero:
[tex]\[ x^2 + 2x + 1 - 5x - 1 = 0 \implies x^2 - 3x = 0 \][/tex]
- This simplifies to a quadratic equation with [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = 0\)[/tex].
10. [tex]\((x + 2)^3 = x(x^2 - 10x + 25)\)[/tex]
- Expand and simplify both sides:
[tex]\[ (x + 2)^3 \implies x^3 + 6x^2 + 12x + 8 \][/tex]
[tex]\[ x(x^2 - 10x + 25) \implies x^3 - 10x^2 + 25x \][/tex]
- Equate and simplify:
[tex]\[ x^3 + 6x^2 + 12x + 8 = x^3 - 10x^2 + 25x \][/tex]
[tex]\[ 16x^2 - 13x + 8 = 0 \][/tex]
- This is a quadratic equation with [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex].
Summarizing the quadratic equations and their coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
1. [tex]\((x + 2)(x - 3) = 5\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], [tex]\(c = -11\)[/tex]
2. [tex]\(\left(x^2 + 1 \times 10\right) = 0\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = 0\)[/tex], [tex]\(c = 10\)[/tex]
3. [tex]\(4(x + 1)^2 = 2(x - 1)\)[/tex]
- [tex]\(a = 4\)[/tex], [tex]\(b = 6\)[/tex], [tex]\(c = 6\)[/tex]
4. [tex]\(x^2 + 2x + 1 = 5x + 1\)[/tex]
- [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], [tex]\(c = 0\)[/tex]
5. [tex]\((x + 2)^3 = x(x^2 - 10x + 25)\)[/tex]
- [tex]\(a = 16\)[/tex], [tex]\(b = -13\)[/tex], [tex]\(c = 8\)[/tex]
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.
