Final answer:
The solution involves solving a quadratic equation using the quadratic formula, multiplying binomials, finding the area of a rectangle expressed with polynomials, and factoring a difference of squares.
Step-by-step explanation:
Real-life application problems that involve multiplying polynomials are common in various fields such as engineering, economics, and physical sciences. Here are the solutions to the parts of the question:
- Solving the equation 2x^2 + 3x - 5 = 0 can be done using the quadratic formula. This is a mathematical function known as a second-order polynomial or quadratic function:
- Multiplying binomials like (x + 2)(x - 3) involves applying the distributive property, also known as the FOIL method in this case.
- To find the area of a rectangular garden with sides represented by the polynomials (2x + 1) and (3x - 2), you multiply the two expressions, which is another example of polynomial multiplication.
- Factoring the expression 4x^2 - 9 involves recognizing it as a difference of squares and applying the appropriate factoring technique.
Now, let's solve each part in detail:
Solve the quadratic equation 2x^2 + 3x - 5 = 0 using the quadratic formula:
x =
[-b ± √(b^2 - 4ac)] / (2a)
Here, a = 2, b = 3, and c = -5. Plug these values into the formula to find the values of x.
Multiply (x + 2)(x - 3) by expanding the expression:
(x + 2)(x - 3) = x^2 - 3x + 2x - 6
Simplify it to:
x^2 - x - 6
For the rectangular garden, multiply (2x + 1) by (3x - 2) to get the area:
A = (2x + 1)(3x - 2)
Expand it to:
A = 6x^2 - 4x + 3x - 2
Simplify it to:
A = 6x^2 - x - 2
Factor the expression 4x^2 - 9 by recognizing the pattern a^2 - b^2 = (a + b)(a - b):
4x^2 - 9 = (2x)^2 - (3)^2
Factor it into:
(2x + 3)(2x - 3)