Answer :
Sure! Let's solve each part of the worksheet step by step.
### PART A
#### 1. Define Quadratic Equation.
A quadratic equation is a second-order polynomial equation in a single variable [tex]\(x\)[/tex] with the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a \neq 0\)[/tex]. The general form includes [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex] as constants, with [tex]\(x\)[/tex] as the variable raised to the power of 2.
#### 2. How are quadratic equations different from other equations?
Quadratic equations are different from linear equations in that they include a term with [tex]\(x^2\)[/tex]. This [tex]\(x^2\)[/tex] term means the graph of a quadratic equation is a parabola, while the graph of a linear equation is a straight line. Also, quadratic equations can have two real solutions, one real solution, or no real solutions, whereas linear equations always have exactly one solution.
#### 3. What is the standard form of quadratic equations?
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex] and [tex]\(a \neq 0\)[/tex],
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex],
- [tex]\(c\)[/tex] is the constant term.
#### 4. What are the methods in solving quadratic equations?
The methods for solving quadratic equations include:
- Factoring,
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex],
- Completing the square,
- Graphing.
### PART B
Let's identify which of the given equations are quadratic and find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] if applicable.
#### 1. [tex]\(12 + 3x^2 = 9x - 4\)[/tex]
Rearrange the equation to standard form:
[tex]\[3x^2 - 9x + 16 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 3, \, b = -9, \, c = 16\][/tex]
#### 2. [tex]\(x(x + 2) = x^2 + 4\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 + 2x = x^2 + 4\][/tex]
[tex]\[2x - 4 = 0\][/tex]
This simplifies to a linear equation [tex]\(2x - 4 = 0\)[/tex], which is not a quadratic equation.
#### 3. [tex]\((x + 2)(x + 5) = 0\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 + 7x + 10 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 1, \, b = 7, \, c = 10\][/tex]
#### 4. [tex]\((x - 3)^2 + 10 = 0\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 - 6x + 9 + 10 = 0\][/tex]
[tex]\[x^2 - 6x + 19 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 1, \, b = -6, \, c = 19\][/tex]
### Summary of Part B
- Equation 1: Quadratic, [tex]\(a = 3\)[/tex], [tex]\(b = -9\)[/tex], [tex]\(c = 16\)[/tex]
- Equation 2: Not quadratic
- Equation 3: Quadratic, [tex]\(a = 1\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = 10\)[/tex]
- Equation 4: Quadratic, [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 19\)[/tex]
With these detailed solutions and explanations, you should have a clear understanding of how to identify and work with quadratic equations.
### PART A
#### 1. Define Quadratic Equation.
A quadratic equation is a second-order polynomial equation in a single variable [tex]\(x\)[/tex] with the form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a \neq 0\)[/tex]. The general form includes [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex] as constants, with [tex]\(x\)[/tex] as the variable raised to the power of 2.
#### 2. How are quadratic equations different from other equations?
Quadratic equations are different from linear equations in that they include a term with [tex]\(x^2\)[/tex]. This [tex]\(x^2\)[/tex] term means the graph of a quadratic equation is a parabola, while the graph of a linear equation is a straight line. Also, quadratic equations can have two real solutions, one real solution, or no real solutions, whereas linear equations always have exactly one solution.
#### 3. What is the standard form of quadratic equations?
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where:
- [tex]\(a\)[/tex] is the coefficient of [tex]\(x^2\)[/tex] and [tex]\(a \neq 0\)[/tex],
- [tex]\(b\)[/tex] is the coefficient of [tex]\(x\)[/tex],
- [tex]\(c\)[/tex] is the constant term.
#### 4. What are the methods in solving quadratic equations?
The methods for solving quadratic equations include:
- Factoring,
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex],
- Completing the square,
- Graphing.
### PART B
Let's identify which of the given equations are quadratic and find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] if applicable.
#### 1. [tex]\(12 + 3x^2 = 9x - 4\)[/tex]
Rearrange the equation to standard form:
[tex]\[3x^2 - 9x + 16 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 3, \, b = -9, \, c = 16\][/tex]
#### 2. [tex]\(x(x + 2) = x^2 + 4\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 + 2x = x^2 + 4\][/tex]
[tex]\[2x - 4 = 0\][/tex]
This simplifies to a linear equation [tex]\(2x - 4 = 0\)[/tex], which is not a quadratic equation.
#### 3. [tex]\((x + 2)(x + 5) = 0\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 + 7x + 10 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 1, \, b = 7, \, c = 10\][/tex]
#### 4. [tex]\((x - 3)^2 + 10 = 0\)[/tex]
Expand and rearrange the equation:
[tex]\[x^2 - 6x + 9 + 10 = 0\][/tex]
[tex]\[x^2 - 6x + 19 = 0\][/tex]
This is a quadratic equation. The values are:
[tex]\[a = 1, \, b = -6, \, c = 19\][/tex]
### Summary of Part B
- Equation 1: Quadratic, [tex]\(a = 3\)[/tex], [tex]\(b = -9\)[/tex], [tex]\(c = 16\)[/tex]
- Equation 2: Not quadratic
- Equation 3: Quadratic, [tex]\(a = 1\)[/tex], [tex]\(b = 7\)[/tex], [tex]\(c = 10\)[/tex]
- Equation 4: Quadratic, [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], [tex]\(c = 19\)[/tex]
With these detailed solutions and explanations, you should have a clear understanding of how to identify and work with quadratic equations.
