Problem 26 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ 3 y^{2}+2 y-1=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation using the quadratic formula: $3y^2 + 2y - 1 = 0$ Answer: The two solutions for the given quadratic equation are $y = \frac{1}{3}$ and $y = -1$.

Step 1: Identify a, b, and c

In the given equation, we have: $$ 3y^2 + 2y - 1 = 0 $$ Compare this with the standard quadratic equation form: $$ ay^2 + by + c = 0 $$ We can identify the values of a, b, and c: $$ a = 3, \quad b = 2, \quad c = -1 $$

Step 2: Apply the quadratic formula

Now, apply the quadratic formula with the given values of a, b, and c: $$ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Substitute the values a = 3, b = 2, and c = -1: $$ y = \frac{-2 \pm \sqrt{2^2 - 4(3)(-1)}}{2(3)} $$

Step 3: Simplify the equation

Next, simplify the equation and solve for y: $$ y = \frac{-2 \pm \sqrt{4 + 12}}{6} $$ $$ y = \frac{-2 \pm \sqrt{16}}{6} $$ $$ y = \frac{-2 \pm 4}{6} $$

Step 4: Find two solutions for y

Now, find the two possible solutions for y using the plus and minus signs in the formula: 1. Using the plus sign: $$ y = \frac{-2 + 4}{6} \Rightarrow y = \frac{2}{6} \Rightarrow y = \frac{1}{3} $$ 2. Using the minus sign: $$ y = \frac{-2 - 4}{6} \Rightarrow y = \frac{-6}{6} \Rightarrow y = -1 $$ Hence, the two solutions for the given quadratic equation are: $$ y = \frac{1}{3} \quad \text{and} \quad y = -1 $$

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental concept in algebra, representing equations of the form \( ax^2 + bx + c = 0 \) where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). These equations involve the variable \( x \) raised to the second degree, or squared.

Characteristics of a quadratic equation:

• The graph of a quadratic equation is a parabola.
• The solutions of a quadratic equation correspond to the points where the parabola intersects the x-axis, also known as the equation's roots.
• Quadratic equations can have two, one, or no real solutions depending on the discriminant, \( b^2 - 4ac \).

The standard procedure for solving these equations involves various methods such as factoring, completing the square, or using the quadratic formula. Understanding these basics helps when dealing with more complex algebraic expressions and solving problems effectively.

Solving Equations

Solving equations is a core practice in algebra, involving the process of finding unknown values that satisfy the equation. With quadratic equations, the goal is to determine the solutions for \( y \) or \( x \) using specific techniques.

When we tackle a problem like \( 3y^2 + 2y - 1 = 0 \), using the quadratic formula is a systematic approach. Here's the general idea:

• Identify the coefficients \( a \), \( b \), and \( c \) in the quadratic equation.
• Apply the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots.
• Calculate the discriminant, \( b^2 - 4ac \), which determines the nature of the roots.
• Solve for the variable to find possible solutions.

Checking each step carefully is essential for accuracy in complex problems. Breaking down each part ensures one doesn't overlook minor errors that could alter the outcome.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols to tackle various problems. It serves as a building block for higher-level math and science subjects.

In algebra, understanding quadratic equations and their solutions is key. This branch uses symbols to represent numbers in formulae and equations, making it possible to solve for unknown quantities. Here's why it’s valuable:

• Promotes problem-solving skills by providing a framework to develop logical reasoning and critical thinking.
• Forms the basis for further study in calculus and other advanced topics.
• Has practical applications in physics, engineering, statistics, and economics.

By mastering algebraic concepts like solving quadratic equations, students enhance their ability to approach complex mathematical problems confidently and systematically. Learning algebra helps in developing a strong mathematical foundation.