Problem 26 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$t 2-8 t-2=0$$

Step-by-Step Solution

Answer

Solutions are \( t = 4 + 3\sqrt{2} \) and \( t = 4 - 3\sqrt{2} \).

Step 1: Identify the Quadratic Equation Form

The given equation is \( t^2 - 8t - 2 = 0 \). Identify the coefficients \( a \), \( b \), and \( c \) from the standard quadratic equation form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -8 \), \( c = -2 \).

Step 2: Write the Quadratic Formula

The quadratic formula is written as \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides the solutions for any quadratic equation.

Step 3: Calculate the Discriminant

The discriminant of a quadratic equation is given by \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \) to find it: \( (-8)^2 - 4(1)(-2) = 64 + 8 = 72 \).

Step 4: Apply the Quadratic Formula - Positive Solution

Now substitute the values into the quadratic formula for both the plus and minus: \( t = \frac{8 \pm \sqrt{72}}{2} \). First, calculate the positive root: \( t = \frac{8 + \sqrt{72}}{2} \). Simplify \( \sqrt{72} \) to \( 6\sqrt{2} \). Therefore, \( t = \frac{8 + 6\sqrt{2}}{2} \ = 4 + 3\sqrt{2} \).

Step 5: Apply the Quadratic Formula - Negative Solution

Now calculate the negative root: \( t = \frac{8 - \sqrt{72}}{2} \ = \frac{8 - 6\sqrt{2}}{2} \ = 4 - 3\sqrt{2} \).

Step 6: Write the Solutions of the Quadratic Equation

The solutions to the equation \( t^2 - 8t - 2 = 0 \) using the quadratic formula are \( t = 4 + 3\sqrt{2} \) and \( t = 4 - 3\sqrt{2} \).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). When solving quadratic equations, the goal is to find the values of \( x \) that satisfy the equation.
Our specific equation given here is \( t^2 - 8t - 2 = 0 \). In this case:

• \( a = 1 \)
• \( b = -8 \)
• \( c = -2 \)

Notice how the equation is already in the standard form of a quadratic, making it easier to identify these coefficients directly.
This classification as a quadratic equation lets us use powerful tools like the quadratic formula to find solutions or roots of the equation.

Discriminant

The discriminant is an important component of the quadratic formula, playing a critical role in determining the nature of the roots for a quadratic equation. It is given by the expression \( b^2 - 4ac \).
For the equation \( t^2 - 8t - 2 = 0 \):

• \( b^2 = (-8)^2 = 64 \)
• \( 4ac = 4 \times 1 \times (-2) = -8 \)

Therefore, the discriminant \( b^2 - 4ac = 64 + 8 = 72 \).
The value of the discriminant provides insights:

• If it is positive, the quadratic equation has two distinct real roots.
• If it is zero, it means there is one real root (a repeated real root).
• If it is negative, the roots are complex and imaginary.

Since our discriminant is positive, we expect two distinct real roots for our equation.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) (or any variable used, like \( t \) in our case) that make the equation true. Solving a quadratic equation means finding these roots.
The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) allows us to directly compute the roots. Here, "\( \pm \)" indicates there will be two values giving us two solutions.
Using our specific case with \( a = 1 \), \( b = -8 \), and \( c = -2 \), the formula becomes \( t = \frac{8 \pm \sqrt{72}}{2} \):

• For the positive value, we simplify it to \( t = 4 + 3\sqrt{2} \).
• For the negative value, it simplifies to \( t = 4 - 3\sqrt{2} \).

These values \( 4 + 3\sqrt{2} \) and \( 4 - 3\sqrt{2} \) are the roots of our quadratic equation, representing the solutions where \( t \) satisfies \( t^2 - 8t - 2 = 0 \). Each root represents a point where the quadratic curve intersects the \( t \)-axis on a graph.