Problem 26 - Precalculus : Building Concepts and Connections

Question

Factor to find the \(x\)-intercepts of the parabola described by the quadratic function. Also find the real zeros of the function. $$h(t)=-3 t^{2}+10 t-8$$

Step-by-Step Solution

Answer

The x-intercepts of the function, which are also the real zeros, are \(t=-4\) and \(t=\frac{2}{3}\).

Step 1: Identify the Quadratic Function

The given equation is a quadratic function and it is \(h(t)=-3 t^{2}+10 t-8\). This equation is in the standard form of a quadratic function, which is \(ax^2 + bx + c = 0\). Here, \(a=-3\), \(b=10\), and \(c=-8\).

Step 2: Factor the Quadratic Function

To factor the quadratic function, we are looking for two numbers that multiply to give the product \(ac=-24\) (which is \(-3 \times -8\)) and add to give \(b=10\). The numbers that satisfy these conditions are -2 and 12. Hence, we can rewrite the middle term as the sum of the terms -2t and 12t. We get: \(-3t^2 - 2t + 12t - 8 = 0\). We can factor by grouping, giving us: \(t(-3t+2)+4(-3t+2)=0\). This simplifies to: \((t+4)(-3t+2)=0\)

Step 3: Set Each Factor to Zero

In order to find the x-intercepts or the real zeros of the function, you set each factor equal to zero and solve for \(t\). Thus we get: \(t+4=0\) or \(-3t +2=0\). Solving these gives \(t=-4\) and \(t=\frac{2}{3}\).

Key Concepts

Understanding X-Intercepts

X-intercepts are points where a graph crosses the x-axis. At these points, the value of the function is zero. In simpler terms, they are the solutions to the equation when the output, or y-value, is zero. For a quadratic function like \( h(t) = -3t^2 + 10t - 8 \), finding the x-intercepts means solving \( -3t^2 + 10t - 8 = 0 \). When you factor and solve the quadratic, you find the x-intercepts, which tell you where the graph intersects the x-axis.
This means:

• The x-intercepts are the same as the roots or solutions of the equation \( -3t^2 + 10t - 8 = 0 \).
• In this example, the intercepts are \( t = -4 \) and \( t = \frac{2}{3} \).
• These points are significant as they reveal much about the behavior of the function.

Exploring Real Zeros

Real zeros, or real roots, are the values of \( t \) that make the quadratic equation true. In terms of the graph, they represent the same points as the x-intercepts. Real zeros are solutions that can be framed over the set of real numbers, meaning they aren't complex or imaginary.
Here's what you need to know:

• For the quadratic \( h(t) = -3t^2 + 10t - 8 \), finding real zeros involves either factoring or using the quadratic formula.
• Using factorization, like in our example, gives zeros of \( t = -4 \) and \( t = \frac{2}{3} \).
• They are real because they do not result in square roots of negative numbers.
• Real zeros can be checked by substituting back into the original equation to see if they satisfy \( h(t) = 0 \).

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations that are not easily factored. It is given as:\[t = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]This formula provides the solutions for any quadratic equation \( ax^2 + bx + c = 0 \) by substituting the coefficients \( a \), \( b \), and \( c \).
This is how it works:

• Identify \( a \), \( b \), and \( c \) from the equation, which in our case are \( a = -3 \), \( b = 10 \), \( c = -8 \).
• Plug these into the quadratic formula to calculate the roots.
• Calculate the discriminant \( b^2 - 4ac \) to determine the nature of the roots, where a positive value indicates two real solutions.
• If the equation is hard to factor, this formula guarantees a solution, whether rational or irrational.

It's an essential tool for any student dealing with quadratic equations.