Answer :
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]
