As Isla prepares to set off fireworks, she wants to ensure the spectators are a safe distance away. When shot in the air, one firework will follow the path h(x)=−x2+15x , where h(x) is the height of the firework, and x is the horizontal distance it travels in feet. To determine how far the firework will travel before it reaches the ground, Isla sets the height equal to zero, and solves the following quadratic equation: 0=−x2+15x . How far will the firework travel before it reaches the ground?(1 point) Responses −15 feet negative 15 feet 30 feet 30 feet 7.5 feet 7.5 feet 15 feet 15 feet
To determine how far the firework will travel before it reaches the ground, we need to solve the equation:
0 = -x^2 + 15x
To solve this quadratic equation, we can factor it or use the quadratic formula. Factoring the equation, we have:
0 = -x(x - 15)
Setting each factor equal to zero, we get:
x = 0 or x - 15 = 0
From the first factor, x = 0 does not make sense in this context as it represents the starting point of the firework. From the second factor, x - 15 = 0, we find:
x = 15
Therefore, the firework will travel 15 feet before it reaches the ground.
Hyung-eun is solving the quadratic equation x2−4x−12=0 by grouping. Her work is shown below. At which step does Hyung-eun first make an error? x2−4x−12=0 Step1: x2+(−6x)+2x−12=0 Step 2: (x2+(−6x))+(2x−12)=0 Step 3: x(x+(−6))+2(x−6)=0 x(x−6)+2(x−6)=0 Step 4: (x+6)(x−2)=0 Step 5: x+6=0 or x−2=0 x=−6 or x=2 (1 point) Responses Step 1 Step 1 Step 2 Step 2 Step 4 Step 4 Step 3
Hyung-eun first makes an error in Step 3.
In Step 3, she distributes the 2 to both terms in parentheses correctly, but when she combines like terms, she incorrectly writes "2(x-6)" as "2(x-6)" instead of "2(x-6)".
The correct step should be:
x(x-6) + 2(x-6) = 0
Therefore, the error occurs in Step 3.
Which of the following quadratic equations can be solved by grouping?(1 point) Responses x2−4x−8=0 x squared minus 4 x minus 8 equals 0 x2+8x−22=0 x squared plus 8 x minus 22 equals 0 x2−12x+18=0 x squared minus 12 x plus 18 equals 0 x2+10x+21=0
The quadratic equation that can be solved by grouping is:
x2−4x−8=0
(x squared minus 4x minus 8 equals 0)
Solving this equation using grouping can involve breaking the middle term (-4x) into two terms that can be grouped and factored.
To determine how far the firework will travel before it reaches the ground, we need to solve the quadratic equation:
0 = -x^2 + 15x
To solve this equation, we can factor out a common factor of -x:
0 = -x(x - 15)
Setting each factor equal to zero gives us the solutions:
x = 0 and x - 15 = 0
Solving for x, we find:
x = 0 or x = 15
Since we're looking for the distance the firework will travel before it reaches the ground, we take the positive value of x:
Distance traveled = 15 feet
Therefore, the firework will travel 15 feet before it reaches the ground.
To determine how far the firework will travel before it reaches the ground, we need to find the value of x when the height, h(x), is equal to zero.
Given the quadratic equation -x^2 + 15x = 0, we can solve it by factoring or using the quadratic formula.
Factoring:
Setting the equation equal to zero, we can rewrite it as: x(-x + 15) = 0. This means that either x = 0 or -x + 15 = 0.
For x = 0, it indicates that the firework is at the starting point.
For -x + 15 = 0, we can solve for x by adding x to both sides: 15 = x. Therefore, we have x = 15.
So, the firework will travel 15 feet before it reaches the ground.
Hence, the correct answer is 15 feet.