What are the solutions to the equation 3(x – 4)(x + 5) = 0?
To find the solutions to the equation 3(x - 4)(x + 5) = 0, we can use the zero product property. Let's break it down step by step:
Step 1: Apply the distributive property on the left side of the equation:
3(x - 4)(x + 5) = 0
3(x^2 + 5x - 4x - 20) = 0
Step 2: Simplify by combining like terms inside the parentheses:
3(x^2 + x - 20) = 0
Step 3: Multiply each term inside the parentheses by 3:
3x^2 + 3x - 60 = 0
Step 4: Set the equation equal to 0:
3x^2 + 3x - 60 = 0
Step 5: To solve this quadratic equation, we can either factor it or use the quadratic formula. Let's try factoring first:
We need to find two numbers whose product is -60 and whose sum is 3.
After some trial and error, we can factor the equation as:
(3x - 12)(x + 5) = 0
Step 6: Set each factor equal to 0 and solve for x:
3x - 12 = 0 or x + 5 = 0
Step 7: Solve for x in each equation:
3x - 12 = 0
3x = 12
x = 4
x + 5 = 0
x = -5
Step 8: The solutions to the equation 3(x - 4)(x + 5) = 0 are x = 4 and x = -5.
To find the solutions to the equation 3(x – 4)(x + 5) = 0, we can use the zero product property. According to the zero product property, if a product of factors is equal to zero, then at least one of the factors must be equal to zero.
In this equation, we have the product 3(x – 4)(x + 5) = 0. To use the zero product property, we set each factor equal to zero and solve for x:
Setting the first factor equal to zero: x – 4 = 0
Adding 4 to both sides: x = 4
Setting the second factor equal to zero: x + 5 = 0
Subtracting 5 from both sides: x = -5
Therefore, the solutions to the equation 3(x – 4)(x + 5) = 0 are x = 4 and x = -5.
Three hundred yards of fencing is being used to fence in a rectangular garden. The area of the garden is modeled by a quadratic function of the rectangle’s width, A(w). What does the second coordinate of the vertex of the quadratic function. A(w), represent?
the reason we always set the polynomial to zero is that if the product of factors is zero, then at least one of the factors must be zero. So, we have
3 = 0
x-4 = 0
x+5 = 0
Now you can easily see which values of x will make 3(x-4)(x+5) = 0
The vertex of a parabola is at (w,k) where k is the actual maximum area possible.