Problem 65 - Algebra 1

Question

Use the quadratic formula to solve the equation. $$-\frac{2}{3} x^{2}-3 x+1=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(-\frac{2}{3} x^{2}-3 x+1=0\) using the quadratic formula are \(x=\frac{3 + \sqrt{27/3}}{-4/3}\) and \(x=\frac{3 - \sqrt{27/3}}{-4/3}\).

Step 1: Identify Coefficients

First, recognize the given equation \(-\frac{2}{3} x^{2}-3 x+1=0\) is in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). In this equation, \(a = -\frac{2}{3}\), \(b = -3\), and \(c = 1\).

Step 2: Apply the Quadratic Formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the quadratic formula \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). This gives \(\frac{-(-3) \pm \sqrt{(-3)^2-4(-2/3)(1)}}{2(-2/3)}\).

Step 3: Solve under the square root

Calculate the term under the square root. This gives \(\frac{3 \pm \sqrt{9 - (4*-2/3)*1}}{-4/3}\).

Step 4: Simplify the Expression

Further simplifying the expression gives \(\frac{3 \pm \sqrt{9+8/3}}{-4/3\). Continue by simplifying under the root which gives \(\frac{3 \pm \sqrt{27/3}}{-4/3}\). Consequentially, the final solutions are \(x=\frac{3 + \sqrt{27/3}}{-4/3}\) and \(x=\frac{3 - \sqrt{27/3}}{-4/3}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of algebraic expression that includes a variable raised to the second power, or squared. The general form of a quadratic equation is \[ ax^2 + bx + c = 0 \]where:

• \( a \) is the coefficient of the squared term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term.

The highest power of the variable \( x \) in a quadratic equation is always 2. Quadratic equations exhibit a parabolic graph when plotted, opening upwards if \( a > 0 \), or downwards if \( a < 0 \). Understanding the structure of this equation is crucial, as it lays the foundation for solving it using the quadratic formula.

Coefficient Identification

Identifying the coefficients in a quadratic equation is a key step to solving it. Each coefficient corresponds to a specific term in the equation. For instance, in the equation \[ -\frac{2}{3} x^2 - 3x + 1 = 0 \]

• The coefficient \( a \) is \(-\frac{2}{3}\), associated with the quadratic term \( x^2 \).
• The coefficient \( b \) is \(-3\), which is linked to the linear term \( x \).
• The constant \( c \) is \(1\), the part of the equation without a variable attached to it.

Correctly identifying these coefficients is essential because they are directly substituted into the quadratic formula to find the solutions of the equation.

Solving Quadratic Equations

To solve a quadratic equation like \[ -\frac{2}{3} x^2 - 3x + 1 = 0 \]we apply the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula is deep-rooted in algebra and directly solves for \( x \) by incorporating the values of \( a \), \( b \), and \( c \).1. **Insert the coefficients:** Start by replacing \( a \), \( b \), and \( c \) into the formula.2. **Calculate the discriminant:** Compute \( b^2 - 4ac \), known as the discriminant. This determines the nature of the roots (real or complex) and also influences the number of solutions (two, one, or none).3. **Solve under the square root:** Simplify \[ \sqrt{b^2 - 4ac} \].This involves solving any arithmetic operations involved.Following these simple steps allows us to methodically solve for \( x \), providing a deeper understanding and clear solution pathway for any quadratic equation.

Algebraic Expressions

Algebraic expressions like quadratic equations are formed with variables and constants connected through operations like addition, subtraction, multiplication, and division.In the context of quadratic equations, an expression like \[ -\frac{2}{3} x^2 - 3x + 1 \]is a combination of terms and operators:

• \(-\frac{2}{3} x^2\) is the quadratic term, highlighting the variable \( x \) raised to the second power.
• \(-3x\) is the linear term, marking \( x \) raised to the first power.
• \(+1\) is the constant, which remains unassociated with any variable.

Understanding each component of these expressions is vital, as it helps in transforming equations to their standard form, crucial for application of the quadratic formula, and forms the basic building blocks for more complex algebraic manipulations.