Problem 72 - Precalculus : Building Concepts and Connections

Question

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$-6 t^{2}+2 t-\frac{1}{3}=0$$

Step-by-Step Solution

Answer

The given quadratic equation has no real solutions. That implies that the graph of the corresponding quadratic function does not intersect the x-axis at any point.

Step 1: Writing the equation in standard form

The given equation is already in standard form: \(-6t^{2} + 2t - 1/3 = 0\) where \(a = -6\), \(b = 2\) and \(c = -1/3\).

Step 2: Finding the roots using the quadratic formula

The roots of the equation can be found using the quadratic formula: \(t = [-b ± sqrt(b^{2} - 4ac)] / (2a)\). Substitute the values \(a = -6\), \(b = 2\) and \(c = -1/3\) into the formula to find the roots.

Step 3: Solving the equation

Substituting the values into the formula, we get \(t = [-2 ± sqrt((2)^{2} - 4*(-6)*(-1/3))] / (2*(-6))\). This further simplifies to \(t = [-2 ± sqrt(4 - 8)] / -12\), which simplifies to \(t = [-2 ± sqrt(-4)] / -12\). As we can't take the square root of a negative number, the equation has no real roots.

Step 4: Relating the roots to the quadratic function

The roots of a quadratic function are the x-values where the function equals zero. In other words, these are the points where the graph of the function intersects the x-axis. Since this equation has no real roots, the graph of the function \(f(t) = -6t^{2} + 2t - 1/3\) will not intersect the x-axis at any point.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. A quadratic equation generally takes the form \[ax^2 + bx + c = 0\]Here,

• \(a\), \(b\), and \(c\) are constants.
• \(x\) represents an unknown variable.

To find the roots, or solutions, of the equation, you can use the quadratic formula which is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]In this formula, the expression under the square root sign, \(b^2 - 4ac\), is known as the discriminant. Depending on its value, the nature of the roots (whether they are real or complex) can be determined. The quadratic formula is essential when factoring is complicated or impossible.
The roots or zeros of the quadratic will give us insight into where the function crosses the x-axis.

Complex Roots

Sometimes, while using the quadratic formula, you may encounter a situation where the discriminant (\(b^2 - 4ac\)) is negative. This results in a negative number under the square root, leading to what are known as complex roots.

• Complex roots occur in conjugate pairs of the form \(a + bi\) and \(a - bi\), where \(i\) is the imaginary unit.

For our given equation, the discriminant calculation yields \(-4\), indicating that there are no real solutions.
Instead, the solutions are \[t = \frac{{-2 \pm \sqrt{-4}}}{-12} = \frac{-2 \pm 2i}{-12}\] which simplifies the roots to \(\frac{1}{6} + \frac{i}{6}\) and \(\frac{1}{6} - \frac{i}{6}\).These solutions imply that the graph of the quadratic does not touch the x-axis but instead exists entirely above or below it.

Standard Form

Quadratic equations should often be expressed in their standard form, which is \[ax^2 + bx + c = 0\].Being in this format makes it easier to apply the quadratic formula or other algebraic techniques.
In standard form:

• \(a\) is the coefficient of \(x^2\).
• \(b\) represents the coefficient of \(x\).
• \(c\) is the constant term.

The standard form not only helps in calculation but is also crucial for identifying key features of a quadratic solution, like the axis of symmetry and vertex form later on. For the given equation \(-6t^2 + 2t - \frac{1}{3} = 0\), it's already in the standard form, making it straightforward to identify \(a = -6\), \(b = 2\), and \(c = -\frac{1}{3}\). This clarity is particularly valuable when using the quadratic formula.

Zeros of a Function

The zeros of a function, particularly in a quadratic context, are the points at which the function takes the value zero. These are the solutions, or roots, of the quadratic equation when set equal to zero. They are the x-values where the function intersects or touches the x-axis.
For a typical quadratic function \(f(x) = ax^2 + bx + c\),

• Zeros can be found by solving \(ax^2 + bx + c = 0\).

In our specific case \(f(t) = -6t^2 + 2t - \frac{1}{3}\), the function's graph doesn't intersect the x-axis since it has complex roots. The absence of real zeros means \(f(t)\) will have either a minimum or maximum value depending on the parabola's direction but will not cross the axis.
The zeroes are vital as they show potential equilibrium points in physical systems modeled by quadratic equations, and graphically, they dictate the parabola’s cut points with the x-axis.