Problem 27 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 8 t=5+2 t^{2} $$

Step-by-Step Solution

Answer

So the solutions to the given quadratic equation are \(t = 2 + \sqrt{6}\) and \(t = 2 - \sqrt{6}\).

Step 1: Rearrange the equation into standard form

Subtract 8t from both sides to move it to one side and get the equation into standard form. You should get: \(2t^2 - 8t + 5 = 0\). Now, \(a = 2\), \(b = -8\), and \(c = 5\).

Step 2: Substitute the values of a, b, and c into the Quadratic formula

Using the Quadratic formula, \(x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}\), substitute the values of a, b, and c which are 2, -8, 5 respectively. Therefore, the equation will be \(t = \frac{-(-8) ± \sqrt{(-8)^2 - 4*2*5}}{2*2}\).

Step 3: Simplify the equation

Simplify the equation into \(t = \frac{8 ± \sqrt{64 - 40}}{4}\). Further simplifying gives \(t = \frac{8 ± \sqrt{24}}{4}\). Then simplify in detail to get \(t = 2 ± \sqrt{6}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form of ax^2 + bx + c = 0. This means it involves terms up to t squared, where the highest power of t is 2. In simpler terms, a quadratic equation is a way to describe a parabola on a graph. There are several ways to solve a quadratic equation, such as by

• factoring
• completing the square
• using the Quadratic Formula

These methods are tools that help in finding the values of t, known as the "roots", which make the equation true. In our example equation 8t = 5 + 2t², it is first necessary to rearrange it into the standard form, so you can easily apply the Quadratic Formula.

Standard Form

The standard form of a quadratic equation is ax^2 + bx + c = 0. It is essential to rearrange a given quadratic equation into this format before solving it. This helps to identify the values of 'a', 'b', and 'c', which are crucial when using the Quadratic Formula. Let's look at the exercise:
Original equation: 8t = 5 + 2t²
Reorganize to get:
2t² - 8t + 5 = 0
Here, we subtracted 8t from both sides to put it in standard form.

• 'a' is the coefficient of t², which is 2.
• 'b' is the coefficient of t, which is -8.
• 'c' is the constant term, which is 5.

Proper organization into standard form paves the way for easy application of solving methods like the Quadratic Formula.

Discriminant

The discriminant is a key component of the Quadratic Formula, providing valuable information about the nature of the solutions. It is given by the part of the formula under the square root: b^2 - 4ac. The discriminant helps us understand whether the roots of a quadratic equation are real or complex:

• If b^2 - 4ac > 0, there are two distinct real roots.
• If b^2 - 4ac = 0, there is exactly one real root.
• If b^2 - 4ac < 0, the roots are complex (not real).

In the solution of our equation, we calculated the discriminant to be 64 - 40 = 24. Since 24 is greater than zero, the quadratic equation has two distinct real solutions. Understanding the discriminant's role gives insight into your equation's solution even before calculator-assisted calculations are made.