Problem 58 - Precalculus : Building Concepts and Connections

Question

Compute the zeros of the quadratic function. $$g(t)=-5 t^{2}+2 t-3$$

Step-by-Step Solution

Answer

The zeros of the function \(g(t) = -5t^2 +2t -3\) are \(t_{1} = 0.2 - 0.2i\sqrt{14}\) and \(t_{2} = 0.2 + 0.2i\sqrt{14}\), which are complex numbers.

Step 1: Identify the coefficients of the quadratic equation

Here the equation is \(g(t) = -5t^2 + 2t - 3\). In a quadratic equation of the form \(ax^2 + bx + c\), \(a\) is the coefficient of \(t^2\), \(b\) is the coefficient of \(t\) and \(c\) is the constant term. So in this exercise \(a = -5\), \(b = 2\), and \(c = -3\).

Step 2: Substitute the coefficients into the quadratic formula

We insert our identified coefficients \(-5\), \(2\), and \(-3\) into the quadratic formula: \[t = \frac{-2 \pm \sqrt{(2)^2 - 4*(-5)*(-3)}}{2*(-5)}\]

Step 3: Simplify the equation

Next, we have to simplify this expression. First, calculate the term under the square root sign (\(\Delta\)): \[(2)^2 - 4*(-5)*(-3) = 4 - 60 = -56\]. Since \(\Delta\) is negative, the roots of the quadratic equation are complex.

Step 4: Calculate the roots of the equation

Our two roots are \[t_{1}= \frac{-2 + \sqrt{-56}}{-10} = 0.2 - 0.2i\sqrt{14}\] and \[t_{2}= \frac{-2 - \sqrt{-56}}{-10} = 0.2 + 0.2i\sqrt{14}\].

Key Concepts

Complex Roots

When solving quadratic equations, sometimes the solutions are not real numbers but complex numbers. This happens when you encounter a negative number under the square root in the quadratic formula. This part under the square root is called the discriminant, denoted by \( \Delta \).

In our example, the discriminant was \(4 - 60 = -56 \). Since \( \Delta \) is negative, it indicates that the roots are complex rather than real.

Complex roots come in pairs and are expressed in the form \( a \pm bi \), where \( a \) is the real part and \( b \) is the imaginary part. The imaginary unit \( i \) satisfies \( i^2 = -1 \). For \( g(t) = -5t^2 + 2t - 3 \), we calculated the roots:

• \( t_1 = 0.2 - 0.2i \sqrt{14} \)
• \( t_2 = 0.2 + 0.2i \sqrt{14} \)

These complex numbers show that the equation does not intersect the x-axis on a graph, highlighting the nature of complex solutions.

Quadratic Formula

The quadratic formula is a universal method used to find the roots of any quadratic equation. It's particularly helpful when the equation does not factor easily. The formula is given as:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

It derives from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Using the quadratic formula helps in determining the nature of the roots based on the discriminant \( \Delta = b^2 - 4ac \).

• When \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the roots are real and identical.
• For \( \Delta < 0 \), the roots are complex and conjugate.

By substituting our coefficients: \( a = -5, b = 2, c = -3 \) into the formula, we calculated the complex roots for the quadratic function \( g(t) = -5t^2 + 2t - 3 \). This process not only assists in solving for \( t \), but also in understanding the behavior of the quadratic graph.

Coefficients Identification

Understanding how to identify and use coefficients in a quadratic equation is key to solving it. Every quadratic equation is generally expressed in the form \( ax^2 + bx + c = 0 \). Here:

• \(a\) is the coefficient of \(x^2\), which determines the parabola's direction (upward if positive, downward if negative).
• \(b\) is the coefficient of \(x\), which affects the axis of symmetry and the location of the vertex of the parabola.
• \(c\) is the constant term, which decides where the parabola intersects the y-axis.

In our example equation \( g(t) = -5t^2 + 2t - 3 \):

• \(a = -5\) means the parabola opens downward.
• \(b = 2\) impacts the symmetry.
• \(c = -3\) indicates the y-intercept is at \(-3\).

Accurately identifying \( a, b, \) and \( c \) allows you to correctly apply the quadratic formula and understand how the quadratic function behaves on a graph.