How would you factor the problem:
7-23r+6r to the second power (only 6r is to the second power)
And if a length of a rectangle i 3 more than twice the width and the are is 90 cm squared than what are the dimensions of the rectangle?
Oh its not rectangle i its rectangle IS sorry for the error. PLEASE HELP!
To factor the expression 7-23r+6r^2, we can look for common factors and use the distributive property. Here's how you can do it step by step:
1. Rearrange the terms in descending order of the exponents: 6r^2 - 23r + 7.
2. Look for pairs of numbers whose product equals the product of the leading coefficient (6) and the constant term (7). In this case, the pair is (1,7) because 1 * 7 = 6.
3. Split the middle term (-23r) using the pair identified in the previous step. Rewrite -23r as -2r - 21r (splitting -23r using -2r and -21r).
4. Factor by grouping:
Group the first two terms and the last two terms:
(6r^2 - 2r) + (-21r + 7).
5. Factor out the greatest common factor from each group:
2r(3r - 1) - 7(3r - 1).
6. Notice that (3r - 1) is a common factor for both groups.
7. Factor out (3r - 1):
(2r - 7)(3r - 1).
Thus, the factored form of 7-23r+6r^2 is (2r - 7)(3r - 1).
Now, let's solve the problem about the rectangle's dimensions:
Let's assume the width of the rectangle is W cm.
According to the problem, the length is 3 more than twice the width, so the length is (2W + 3) cm.
The area of a rectangle is given by the formula: Area = length * width.
So, we can write the equation as:
(2W + 3) * W = 90 cm^2.
To solve this equation for W, you can follow these steps:
1. Distribute: 2W^2 + 3W = 90.
2. Rearrange the equation: 2W^2 + 3W - 90 = 0.
3. Try to factor the quadratic equation. In this case, it's not immediately factorable, so we'll use the quadratic formula.
4. The quadratic formula is: W = (-b ± sqrt(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
For this equation, a = 2, b = 3, and c = -90.
5. Substitute those values into the quadratic formula and calculate W.
Using the quadratic formula, you will get two possible solutions for W: W1 and W2.
Plug these values into (2W + 3) to get the corresponding lengths of the rectangle: (2W1 + 3) and (2W2 + 3).
Therefore, the dimensions of the rectangle will be given by the values of width and length obtained from solving the quadratic equation.