Analyze and solve the following word problems involving quadratic equations and rational algebraic equations below .
1. If the same number is added to the numerator and denominator of-4/7, the result is-9/2.
2. A length of a rectangular parking lot is 36m longer than its width . The area of the parking lot is 5,152m²
(-4 + n) / (7 + n) = -9 / 2
"cross" multiplying ... 2 (-4 + n) = -9 (7 + n) ... 2 n - 8 = - 9 n - 63 ... 11 n = - 55
w * (w + 36) = 5152 ... w^2 + 36 w - 5152 = 0
... use the quadratic formula to find w
I don't understand math like the way I feel for you bye:>:>
To solve the given word problems involving quadratic equations and rational algebraic equations, we will break down each problem and explain the steps to arrive at the solution.
1. If the same number is added to the numerator and denominator of -4/7, the result is -9/2.
Let's represent the number as 'x'. The given equation can be written as:
(-4 + x) / (7 + x) = -9/2
To solve this equation, we will cross-multiply:
-2(-4 + x) = -9(7 + x)
8 - 2x = -63 - 9x
Group the x terms on one side and constants on the other side:
-2x + 9x = -63 - 8
7x = -71
Divide both sides by 7 to solve for x:
x = -71/7
Simplifying the fraction, we find:
x = -10 1/7
So, the number is -10 1/7.
2. A length of a rectangular parking lot is 36m longer than its width. The area of the parking lot is 5,152m².
Let's represent the width of the parking lot as 'x' meters. Since the length is 36 meters longer than the width, the length can be represented as 'x + 36' meters.
The area of a rectangle is given by multiplying its length and width. Therefore, we have:
Length * Width = Area
(x + 36) * x = 5152
Expanding the equation:
x^2 + 36x = 5152
Rearranging the equation to bring it to quadratic form:
x^2 + 36x - 5152 = 0
Now, we have a quadratic equation. To solve this equation, we can either factorize it or use the quadratic formula.
Since the quadratic equation is not easily factorizable, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = 36, and c = -5152. Substituting these values:
x = (-36 ± √(36^2 - 4(1)(-5152))) / (2 * 1)
Simplifying:
x = (-36 ± √(1296 + 20608)) / 2
x = (-36 ± √(21904)) / 2
x = (-36 ± 148) / 2
Now we have two possible values of x:
1. x = (-36 + 148) / 2 = 112 / 2 = 56
2. x = (-36 - 148) / 2 = -184 / 2 = -92
Since we are dealing with dimensions of a parking lot, a negative value for width doesn't make sense. Therefore, our solution is:
Width (x) = 56 meters
Length = Width + 36 = 56 + 36 = 92 meters
Hence, the width of the parking lot is 56 meters, and the length is 92 meters.