(4x-3)^2 = 16x^2 - 24x + 9
You have -9. A perfect square will never end in a negative number.
A: ?
Determine whether the trinomial is a perfect-square. If so, factor. If not, explain.
2. 16x^2 - 24x - 9
A: ?
(I think this trinomial may be a perfect-square, but my textbook tells me otherwise.
You have -9. A perfect square will never end in a negative number.
Step 1: Find the amount of orange juice in the original mixture.
- The original mixture consists of 8 cups of fruit punch, which is composed of 25% orange juice.
- So, the amount of orange juice in the original mixture is 8 cups * 0.25 = 2 cups.
Step 2: Calculate the desired amount of orange juice in the final mixture.
- We want the final mixture to have a concentration of 40% orange juice.
- Let's assume Maria adds x cups of orange juice to the mixture.
- So, the desired amount of orange juice in the final mixture is (2 cups + x cups).
Step 3: Set up the equation.
- We know that the desired amount of orange juice is (2 cups + x cups), and this should constitute 40% of the total volume of the final mixture.
- Thus, the equation can be written as: (2 cups + x cups) = 0.4 * (8 cups + x cups).
- Simplifying the equation:
2 cups + x cups = 3.2 cups + 0.4x cups.
Step 4: Solve for x.
- Subtracting 0.4x cups from both sides:
2 cups + x cups - 0.4x cups = 3.2 cups.
- Simplifying the equation further:
0.6x cups = 1.2 cups.
- Dividing both sides by 0.6:
x cups = 1.2 cups / 0.6.
- Calculating:
x cups = 2 cups.
Therefore, Maria should add 2 cups of orange juice to the mixture.
2. To determine whether the trinomial 16x^2 - 24x - 9 is a perfect square, we can follow these steps:
Step 1: Check the coefficient of the terms.
- The coefficient of the x^2 term is 16, which is a perfect square since √16 = 4.
- The coefficient of the x term, -24, is not a perfect square.
Step 2: Look at the constant term.
- The constant term in this case is -9, which is also not a perfect square.
Therefore, the trinomial 16x^2 - 24x - 9 is not a perfect square.