Problem 8 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$3 y^{2}-2 y+1=0$$

Step-by-Step Solution

Answer

No real solution; use the quadratic formula to find complex roots.

Step 1: Identify the Equation Type

The equation provided is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 3 \), \( b = -2 \), and \( c = 1 \).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

No real solution; use the quadratic formula to find complex roots.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. A quadratic equation is generally written in the form \( ax^2 + bx + c = 0 \). To find the roots or solutions of the equation, we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula will give you two solutions (or roots), which can be real or complex. The expression \( b^2 - 4ac \) is known as the discriminant. It can tell us a lot about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the roots are complex and not real numbers.

Using the quadratic formula is often ideal when factoring methods seem difficult or impossible. Just plug in the values of \( a \), \( b \), and \( c \) from your equation into the formula to find your solutions.

Factoring Quadratic Equations

Factoring is another common method for solving quadratic equations. It involves rewriting the equation as a product of two binomials set equal to zero. This method works well when the roots of the quadratic are rational numbers.Start by finding two numbers that multiply to \( ac \) (where \( a \) and \( c \) are coefficients from \( ax^2 + bx + c \)) and add up to \( b \). This can be a bit of a trial and error process. Once you have found these numbers, you can rewrite the middle term \( bx \) using them, factor by grouping, and then apply the zero-product property to solve for the roots.For example, consider \( 2y^2 + 5y + 3 = 0 \). We look for two numbers that multiply to \( 2 \times 3 = 6 \) and add to \( 5 \). The numbers \( 2 \) and \( 3 \) work, so:\[2y^2 + 2y + 3y + 3 = 0 \]We can group and factor:\[2y(y + 1) + 3(y + 1) = 0 \]Thus, the equation becomes:\[(2y + 3)(y + 1) = 0 \]The solutions can be found by setting each factor equal to zero: \( 2y + 3 = 0 \) or \( y + 1 = 0 \). Solve these to find \( y \). Factoring is great when it's possible, as it gives an immediate insight into the nature of the solutions.

Completing the Square

Completing the square is a method that can be used to solve any quadratic equation, and it often provides a deeper understanding of quadratic functions. This technique involves rearranging the equation to create a perfect square trinomial, which can then be easily solved.Here’s a step-by-step approach to completing the square:

• Start with equation \( ax^2 + bx + c = 0 \).
• Divide all terms by \( a \), making the coefficient of \( x^2 \) equal to 1.
• Move the constant term \( c/a \) to the other side of the equation.
• Add and subtract \( \left( \frac{b}{2a} \right)^2 \) on the left side to complete the square.
• Rewrite the left side as a binomial squared.
• Solve for \( x \) by taking the square root of both sides and then solving for \( x \).

For example, if we have the equation \( x^2 - 6x + 5 = 0 \), we first move the 5 to the other side: \( x^2 - 6x = -5 \).We then find \( \left( \frac{-6}{2} \right)^2 = 9 \), add it to both sides:\[x^2 - 6x + 9 = 4 \]Rewriting this, we have \( (x - 3)^2 = 4 \), leading to solutions after taking roots: \( x - 3 = \pm 2 \).Thus, \( x = 5 \) or \( x = 1 \). Completing the square helps you to understand the vertex form of a quadratic function and can make graphing simpler.