Problem 19 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$ 4 t^{2}+4 t-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{-1 + \sqrt{2}}{2} \) and \( t = \frac{-1 - \sqrt{2}}{2} \).

Step 1: Identify the Equation

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). For this equation, \( a = 4 \), \( b = 4 \), and \( c = -1 \).

Step 2: Decide on a Solution Method

We will use the quadratic formula to solve this equation since it universally applies to all quadratic equations. The quadratic formula is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate the Discriminant

First, calculate the discriminant \( b^2 - 4ac \). Substitute the values: \( b^2 = 4^2 = 16 \) and \( 4ac = 4 \times 4 \times (-1) = -16 \). Thus, the discriminant is \( 16 + 16 = 32 \).

Step 4: Apply the Quadratic Formula

Now substitute \( a = 4 \), \( b = 4 \), and \( c = -1 \) into the quadratic formula:\[ t = \frac{-4 \pm \sqrt{32}}{2 \times 4} \]

Step 5: Simplify the Solution

Simplify \( \sqrt{32} \) to \( 4\sqrt{2} \) as \( 32 = 16 \times 2 \). So the expression becomes:\[ t = \frac{-4 \pm 4\sqrt{2}}{8} \]This further simplifies to:\[ t = \frac{-1 \pm \sqrt{2}}{2} \]

Step 6: Write the Solutions

The solutions for the equation \( 4t^2+4t-1=0 \) are \( t = \frac{-1 + \sqrt{2}}{2} \) and \( t = \frac{-1 - \sqrt{2}}{2} \).

Key Concepts

Quadratic Formula

When solving quadratic equations, the quadratic formula comes in handy. This formula can solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is written as:

• \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

The variables \( a \), \( b \), and \( c \) are coefficients from the equation:

• \( a \) is the coefficient of \( t^2 \)
• \( b \) is the coefficient of \( t \)
• \( c \) is the constant term

This formula provides two solutions due to the "\( \pm \)" sign, which indicates both addition and subtraction of the square root term. Always substitute the coefficients accurately to get the correct solutions.

Discriminant

The discriminant is part of the quadratic formula, located under the square root sign. Its formula is \( b^2 - 4ac \).
The discriminant is significant because it tells us about the nature of the roots without solving the entire equation:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the roots are complex or imaginary, meaning they are not real numbers.

In our example, the discriminant \( 32 \) indicates that there are two distinct real roots, because it is greater than zero.

Simplifying Radicals

Simplifying radicals is essential when dealing with square roots within the quadratic formula. It helps simplify your solution for clarity. Let's simplify \( \sqrt{32} \) as an example:

• First, find a perfect square factor of 32. Here, 16 is a perfect square since \( 16 \times 2 = 32 \).
• Therefore, \( \sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} \).
• Simplify \( \sqrt{16} \), which equals 4, leaving \( \sqrt{32} = 4\sqrt{2} \).

By simplifying radicals, your answer is neater and more precise, showing you clearly the structure and size of the roots.