Problem 20 - Intermediate Algebra

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$ 5 t^{2}+5 t-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( t = \frac{-5 + 3\sqrt{5}}{10} \) and \( t = \frac{-5 - 3\sqrt{5}}{10} \).

Step 1: Identify the Type of Equation

The equation given is a quadratic equation in the form of \( ax^2 + bx + c = 0 \), where \( a = 5 \), \( b = 5 \), and \( c = -1 \).

Step 2: Choose the Solution Method

In this case, we will use the quadratic formula to solve for \( t \), which is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This method is suitable since we may not easily factorize the equation.

Step 3: Substitute Into the Quadratic Formula

Substitute \( a = 5 \), \( b = 5 \), and \( c = -1 \) into the quadratic formula: \( t = \frac{-5 \pm \sqrt{5^2 - 4 \times 5 \times (-1)}}{2 \times 5} \).

Step 4: Simplify the Expression Under the Square Root

Calculate the value under the square root (the discriminant): \( b^2 - 4ac = 25 + 20 = 45 \).

Step 5: Compute the Square Root and Continue Simplification

The square root of 45 simplifies to \( \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} \). Now, substitute back into the equation: \( t = \frac{-5 \pm 3\sqrt{5}}{10} \).

Step 6: Simplify Further to Find the Solutions

Separate the equation into the two possible solutions for \( t \):1. \( t_1 = \frac{-5 + 3\sqrt{5}}{10} \)2. \( t_2 = \frac{-5 - 3\sqrt{5}}{10} \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used in algebra to solve quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula provides a straightforward method to find the values of \( x \) that satisfy the equation. It is expressed as:

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

This formula works by finding where the parabola described by the quadratic equation crosses the x-axis. The values in the formula—\( a \), \( b \), and \( c \)—are simply the coefficients from the equation. Using the quadratic formula allows you to solve any quadratic equation, even when factoring is not easily possible. By applying this formula, you substitute the coefficients and perform arithmetic operations to find the solutions. It requires calculating the discriminant (the expression under the square root) first, which leads us to our next key concept.

Discriminant Calculation

The discriminant is a fundamental part of the quadratic formula which helps determine the nature and number of the solutions to a quadratic equation. It is denoted by the expression \( b^2 - 4ac \). When solving with the quadratic formula, you start by calculating the discriminant because:

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, which means the parabola touches the x-axis at only one point.
• If the discriminant is negative, there are no real solutions, only complex ones.

In the provided example, the discriminant can be calculated as \( 5^2 - 4 \times 5 \times (-1) = 25 + 20 = 45 \), which is positive. Thus, the equation has two real solutions, and these solutions can be further simplified using the square root of the discriminant, which signals that the solutions are unequal and real.

Quadratic Equations

Quadratic equations form the basis of many algebraic problems and are equations that include a variable squared, such as \( ax^2 + bx + c = 0 \). These equations create a parabola when graphed on a coordinate plane, and the main interest usually lies in finding the points where the parabola intersects the x-axis.
Quadratic equations can sometimes be solved through factoring, completing the square, or by using the quadratic formula.

• Factorizing involves rewriting the quadratic equation as a product of two binomials if possible.
• Completing the square rewrites the equation for easier solving by making it a perfect square trinomial.
• The quadratic formula provides a reliable method to find solutions regardless of the difficulty involved in factoring.

By understanding quadratic equations, you unlock the ability to solve a wide array of algebraic problems, enriching your mathematical toolkit. They are useful in many aspects of science, engineering, and various real-world applications.