Problem 20 - Algebra for College Students

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 Quadratic Equation

The given quadratic equation is \(5t^2 + 5t - 1 = 0\). It is in standard quadratic form \(at^2 + bt + c = 0\) where \(a = 5\), \(b = 5\), and \(c = -1\).

Step 2: Choose a Method to Solve

For this equation, using the quadratic formula is suitable because it can handle any quadratic equation and provides exact solutions. The quadratic formula is \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

The discriminant \(D\) is calculated using \(b^2 - 4ac\). Substitute the values: \(D = 5^2 - 4(5)(-1)\). Simplifying, \(D = 25 + 20 = 45\).

Step 4: Apply the Quadratic Formula

Substitute \(b = 5\), \(a = 5\), and \(D = 45\) into the quadratic formula: \(t = \frac{-5 \pm \sqrt{45}}{2(5)}\). Simplify: \(t = \frac{-5 \pm \sqrt{45}}{10}\).

Step 5: Simplify the Square Root

Simplify \(\sqrt{45}\) as \(\sqrt{9 \times 5} = 3\sqrt{5}\). Substitute back: \(t = \frac{-5 \pm 3\sqrt{5}}{10}\).

Step 6: Simplify the Expression

The solutions can be further simplified: \(t = \frac{-5 + 3\sqrt{5}}{10}\) and \(t = \frac{-5 - 3\sqrt{5}}{10}\). These are the two solutions to the quadratic equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \( at^2 + bt + c = 0 \). The formula itself is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]It works for any quadratic equation, regardless of whether it can be factored easily or not. The "\( \pm \)" symbol in the formula indicates that there are generally two solutions to a quadratic equation, although they might be the same if the discriminant is zero.

• \( a \), \( b \), and \( c \) are coefficients in the equation.
• \( t \) represents the solutions of the quadratic equation.

The quadratic formula is often preferred because it provides exact solutions, making it an excellent choice when accuracy and completeness are necessary.

Discriminant

The discriminant is a key component of the quadratic formula, represented by \( D = b^2 - 4ac \). This part can tell you a lot about the nature of the solutions without actually solving the equation. A positive discriminant indicates two distinct real solutions, a zero discriminant indicates one real solution (also called a repeated root), and a negative discriminant hints at two complex solutions. Here, by calculating \( D = 25 + 20 = 45 \), we confirm that the equation has two real and distinct solutions.

• \( D > 0 \): Two distinct real solutions.
• \( D = 0 \): One real solution.
• \( D < 0 \): No real solutions (complex solutions).

Understanding the discriminant helps in predicting the nature of the roots and streamlines the solving process by preparing one mentally for the expected type of solutions.

Solving Quadratics

Solving quadratic equations means finding the values of \( t \) that satisfy the given quadratic expression. In many cases, the process involves either factoring, using the quadratic formula, or completing the square. The quadratic formula, as used in the given problem, is a go-to method because it can solve any quadratic equation.
When using this method:

• Identify \( a \), \( b \), and \( c \) from the equation.
• Calculate the discriminant \( D \).
• Substitute the values into the quadratic formula.

For \(5t^2 + 5t - 1 = 0\), this resulted in solutions \( t = \frac{-5 + 3\sqrt{5}}{10} \) and \( t = \frac{-5 - 3\sqrt{5}}{10} \). Each solution is found by carefully substituting the coefficients and simplifying step by step.

Square Roots

Square roots often come into play when simplifying solutions derived from the quadratic formula. In this exercise, the square root of the discriminant, \( \sqrt{45} \), was simplified to \( 3\sqrt{5} \). Simplifying square roots can make your solutions more manageable and easier to understand.
To simplify a square root like \( \sqrt{45} \):

• Look for perfect square factors: \( 45 = 9 \times 5 \).
• Rewrite as \( \sqrt{9 \times 5} \).
• Split it into: \( \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \).

Getting comfortable with simplifying square roots is helpful not just for quadratics but for all sorts of algebraic expressions where square roots arise. It ensures that your final answers are as simple and neat as possible.