Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
![**Algebraic Operations and Factoring**
In this section, we will explore polynomial multiplication and factoring.
### Multiplication
1. Multiply the binomials:
\[
(x + 3)(x + 5)
\]
2. Multiply the binomials:
\[
(3x + 1)(4x - 2)
\]
### Factoring
Factor each of the following quadratic expressions:
1. \( x^2 + 6x + 8 \)
2. \( x^2 + 3x - 4 \)
3. \( x^2 - 5x + 6 \)
4. \( x^2 - x - 12 \)
### Problem-Solving
*Find the y-intercept, the x-intercepts, vertex, and graph the following:*
(End of visible text)
### Explanations:
- When multiplying binomials, apply the distributive property (also known as FOIL method for binomials), which stands for First, Outer, Inner, Last terms products.
- To factor quadratic expressions, look for pairs of numbers that multiply to the constant term (the third term in the polynomial) and add up to the coefficient of the middle term (the second term in the polynomial).
- Find the y-intercept by setting \( x = 0 \) and solving for \( y \).
- Find the x-intercepts (also known as roots or zeros) by solving the quadratic equation \( f(x) = 0 \).
- The vertex of a parabola given by \( y = ax^2 + bx + c \) can be found using the vertex formula:
\[
x = -\frac{b}{2a}
\]
Then substitute \( x \) back into the equation to find the y-coordinate of the vertex.
The detailed process allows students to solve and graph quadratics effectively. For any graphical representation, plot these points and the vertex to sketch the parabola.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F50ecada7-73ae-429a-bfd7-3c03fe93abe4%2F095a6394-9429-4356-ba07-c714a96c9046%2Fam1frh6.jpeg&w=3840&q=75)
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