During a water balloon fight, Louisa uses a slingshot to fire a water balloon over the fortified wall of the opponent. When fired from the slingshot, the balloon will follow the path h(x)=−x2+12x , whereh(x) is the height of the balloon in feet, and x is the horizontal distance it will travel. To determine how far the balloon will travel before it reaches the ground, Louisa sets the height equal to zero and solves the quadratic equation 0=−x2+12x . How far will the balloon travel before it reaches the ground?(1 point) The balloon will travel feet before it reaches the ground.

Answer 1

To find out how far the balloon will travel before it reaches the ground, we need to solve the quadratic equation:

\[ 0 = -x^2 + 12x \]

We can factor this equation:

\[ 0 = x(-x + 12) \]

Setting each factor to zero gives us:

1. \( x = 0 \)
2. \( -x + 12 = 0 \) → \( x = 12 \)

The solutions for \( x \) represent the two points where the height \( h(x) \) is zero. The first solution \( x = 0 \) corresponds to the initial position of the water balloon when it is fired, and the second solution \( x = 12 \) represents the horizontal distance traveled before it reaches the ground.

Thus, the balloon will travel:

\[ \text{The balloon will travel } 12 \text{ feet before it reaches the ground.} \]

Answer 2

Which of the following tables shows the correct steps to factor out the GCF and solve the quadratic equation 7x2=56x?(1 point) Responses Put the quadratic in standard form 7x2−56x=0 Factor out the GCF 7x(x−8)=0 Set each factor equal to zero 7x=0 and x−8=0 The solutions are x=−7 and x=8 Put the quadratic in standard form 7 x squared minus 56 x equals 0 Factor out the GCF 7 x left parenthesis x minus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x minus 8 equals 0 The solutions are x equals negative 7 and x equals 8 Put the quadratic in standard form 7x2+56x=0 Factor out the GCF 7x(x+8)=0 Set each factor equal to zero 7x=0 and x+8=0 The solutions are x=−8 and x=0 Put the quadratic in standard form 7 x squared plus 56 x equals 0 Factor out the GCF 7 x left parenthesis x plus 8 right parenthesis equals 0 Set each factor equal to zero 7 x equals 0 and x plus 8 equals 0 The solutions are x equals negative 8 and x equals 0 Put the quadratic in standard form x−8=0 Factor out the GCF No GCF Set each factor equal to zero x−8=0 The solutions are x=8 Put the quadratic in standard form x minus 8 equals 0 Factor out the GCF No GCF Set each factor equal to zero x minus 8 equals 0 The solutions are x equals 8 Put the quadratic in standard form 7x2−56x=0 Factor out the GCF 7x(x−8)=0 Set each factor equal to zero 7x=0 and x−8=0 The solutions are x=0 and x=8

Answer 3

To solve the equation \(7x^2 = 56x\), we start by moving everything to one side to rewrite it in standard form:

1. Put the quadratic in standard form: \[ 7x^2 - 56x = 0 \]

2. Factor out the greatest common factor (GCF): \[ 7x(x - 8) = 0 \]

3. Set each factor equal to zero: \[ 7x = 0 \quad \text{and} \quad x - 8 = 0 \]

4. Solve for x: \[ x = 0 \quad \text{and} \quad x = 8 \]

The correct steps for factoring the quadratic equation and deriving the solutions are:

Put the quadratic in standard form: \(7x^2-56x=0\)
Factor out the GCF: \(7x(x-8)=0\)
Set each factor equal to zero: \(7x=0\) and \(x-8=0\)
The solutions are \(x=0\) and \(x=8\)

This matches the last option in the list provided:

Put the quadratic in standard form \(7x^2 - 56x = 0\)
Factor out the GCF \(7x(x - 8) = 0\)
Set each factor equal to zero \(7x = 0\) and \(x - 8 = 0\)
The solutions are \(x = 0\) and \(x = 8\).