Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ 2 x^{2}-x-1=0 $$

Step-by-Step Solution

Answer

The solutions for the quadratic equation \(2x^2 - x - 1 = 0\) are \(x = 1\) and \(x = -\frac{1}{2}\).

Step 1: Identify the coefficients

In the given equation \(2x^2 - x - 1 = 0\), the values of a, b, and c are as follows: a = 2,b = -1, c = -1.

Step 2: Substitute the coefficients into the Quadratic Formula

Substitute a, b, and c into the Quadratic Formula, \(x = \frac{-(-1) \pm \sqrt{(-1^2)-4*(2)(-1)}}{2*(2)}\). Simplify the expression under the square root first.

Step 3: Simplify the expression

By following the order of operations, simplify under the square root first.\[x = \frac{1 \pm \sqrt{1+8}}{4} = \frac{1 \pm \sqrt{9}}{4}\]The above simplifies to: \[x = \frac{1 \pm 3}{4}\]

Step 4: Solve for x

Finally, solve for x by simplifying the fractions: \[x=\frac{1+3}{4}=1 , x=\frac{1-3}{4}=-\frac{1}{2}\]

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a common requirement in algebra. A standard quadratic equation is represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The Quadratic Formula is a reliable tool for finding the solutions, or "roots," of such equations. The formula is given by:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]This formula allows you to find the values of \(x\) that satisfy the quadratic equation. Here's how it works:

• Start by identifying the coefficients \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the formula.
• Calculate the discriminant \(b^2 - 4ac\), which determines the nature of the roots.
• If the discriminant is positive, you'll have two real distinct roots. If it's zero, both roots are real and equal. A negative discriminant indicates complex roots.
• Finally, solve using the positive and negative options in the formula to find the two solutions.

Using these steps, you can easily solve any quadratic equation efficiently.

Order of Operations

When solving equations, especially quadratics with the Quadratic Formula, following the correct order of operations is critical. The order of operations dictates the sequence in which you should perform calculations.
The universally accepted acronym to remember is PEMDAS:

• Parentheses: Solve anything inside parentheses first.
• Exponents: Address any powers or square roots.
• Multiplication and Division: Perform these from left to right.
• Addition and Subtraction: Also performed from left to right.

In the context of the Quadratic Formula, you initially calculate the discriminant \(b^2 - 4ac\), handling squaring and multiplication before moving on to square roots and division.
Being meticulous about following these operations ensures you accurately simplify complex expressions, preventing potential errors that may arise from skipping or misordering steps.

Simplifying Expressions

Simplifying expressions is an integral part of using the Quadratic Formula. It involves breaking down complex expressions into simpler, more manageable forms, making it easier to solve for unknowns.
While using the Quadratic Formula, here's why simplification is essential:

• When you substitute coefficients into the formula, you'll deal with expressions beneath the square root (the discriminant) and fractions that result from dividing by \(2a\).
• After calculating the discriminant, simplify by taking its square root if it's a perfect square.
• The expression "\(\pm\)" means you'll create two separate solutions, each needing their fractions calculated and simplified separately.
• Simplifying fractions involves dividing the numerator by the denominator carefully.

To further simplify, use factoring when possible, especially if numerator and denominator share common factors. Each step of simplification brings you closer to the final solved values of \(x\). By consistently simplifying, you ensure clarity and accuracy in finding roots.