Problem Analysis:
The problem involves solving a quadratic equation to model the height of a ball thrown into the air. The given quadratic equation is:
\[
h = -3t^2 + 5t + 2
\]
where \( h \) represents the height of the ball in meters above the ground, and \( t \) represents time in seconds.
The task is to:
1. Factor the quadratic equation.
2. Determine the horizontal intercepts (roots) of the equation.
3. Find the vertical intercept (initial height).
---
Step-by-Step Solution:
####
Step 1: Factor the Quadratic Equation
The given quadratic equation is:
\[
h = -3t^2 + 5t + 2
\]
To factor this, we need to rewrite it in the form:
\[
h = a(t - r_1)(t - r_2)
\]
where \( r_1 \) and \( r_2 \) are the roots of the equation.
#####
Factoring Process:
1. Identify the coefficients:
- \( a = -3 \)
- \( b = 5 \)
- \( c = 2 \)
2. Use the method of splitting the middle term (\( 5t \)) into two terms whose product equals \( ac \) and whose sum equals \( b \).
- Calculate \( ac \):
\[
ac = (-3)(2) = -6
\]
- Find two numbers whose product is \( -6 \) and whose sum is \( 5 \). These numbers are \( 6 \) and \( -1 \).
3. Rewrite the middle term \( 5t \) as \( 6t - t \):
\[
h = -3t^2 + 6t - t + 2
\]
4. Group the terms:
\[
h = (-3t^2 + 6t) + (-t + 2)
\]
5. Factor out the greatest common factor (GCF) from each group:
\[
h = -3t(t - 2) - 1(t - 2)
\]
6. Factor out the common binomial factor \( (t - 2) \):
\[
h = (-3t - 1)(t - 2)
\]
Thus, the factored form of the quadratic equation is:
\[
h = (-3t - 1)(t - 2)
\]
---
####
Step 2: Determine the Horizontal Intercepts
The horizontal intercepts occur when \( h = 0 \). Set the factored equation equal to zero:
\[
(-3t - 1)(t - 2) = 0
\]
Using the zero-product property, solve for \( t \):
1. Solve \( -3t - 1 = 0 \):
\[
-3t = 1 \implies t = -\frac{1}{3}
\]
2. Solve \( t - 2 = 0 \):
\[
t = 2
\]
Thus, the horizontal intercepts are:
\[
t = -\frac{1}{3} \quad \text{and} \quad t = 2
\]
---
####
Step 3: Find the Vertical Intercept
The vertical intercept occurs when \( t = 0 \). Substitute \( t = 0 \) into the original quadratic equation:
\[
h = -3(0)^2 + 5(0) + 2
\]
Simplify:
\[
h = 2
\]
Thus, the vertical intercept is:
\[
h = 2
\]
---
Final Answer:
The factored form of the quadratic equation is:
\[
h = (-3t - 1)(t - 2)
\]
The horizontal intercepts are:
\[
t = -\frac{1}{3} \quad \text{and} \quad t = 2
\]
The vertical intercept is:
\[
h = 2
\]
\boxed{h = (-3t - 1)(t - 2), \, t = -\frac{1}{3}, \, t = 2, \, h = 2}
Parent Tip: Review the logic above to help your child master the concept of quadratic applications worksheet.
