Problem 56 - Algebra 2

Question

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 3 x^{2}-9 x+2=0 $$

Step-by-Step Solution

Answer

The exact solutions are \( \frac{9 \pm \sqrt{57}}{6} \).

Step 1: Identify Coefficients

The quadratic equation is in the form \( ax^2 + bx + c = 0 \). Identify the coefficients: \( a = 3 \), \( b = -9 \), \( c = 2 \).

Step 2: Write the Quadratic Formula

The quadratic formula is given by: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the equation.

Step 3: Calculate the Discriminant

Calculate the discriminant (inside the square root): \( b^2 - 4ac = (-9)^2 - 4 \cdot 3 \cdot 2 = 81 - 24 = 57 \).

Step 4: Apply the Quadratic Formula

Substitute \( b = -9 \), \( a = 3 \), and \( \text{Discriminant} = 57 \) into the quadratic formula: \( x = \frac{-(-9) \pm \sqrt{57}}{2 \cdot 3} \).

Step 5: Simplify the Expression

Simplify the expression: \( x = \frac{9 \pm \sqrt{57}}{6} \). This represents the exact solutions for the equation.

Key Concepts

Understanding the Discriminant

The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. It is simply a part of the quadratic formula and appears under the square root sign: - The discriminant is calculated using the expression: \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation.
- If the discriminant is positive, it indicates there are two distinct real roots.
- If it is zero, the equation has exactly one real root, also known as a repeated or double root.
- A negative discriminant means there are no real roots; instead, there are two complex roots.In our exercise, the discriminant is calculated as 57. Since 57 is positive, this tells us that the quadratic equation has two distinct real solutions. Understanding this beforehand can be very helpful when predicting the number of potential solutions you'll encounter.

Decoding the Quadratic Equation

A quadratic equation is a second-order polynomial equation in a single variable \( x \). It generally has the form: - \( ax^2 + bx + c = 0 \).
- Here, \( a \), \( b \), and \( c \) are known as the coefficients of the equation.
- These equations are called 'quadratic' because "quad" refers to the square term \( ax^2 \). The goal is to find the value(s) of \( x \) that make this equation true. Quadratic equations can be solved by various methods such as factoring, completing the square, or using the quadratic formula. In the given exercise, we use the quadratic formula for solving. It is especially useful when factors are not readily apparent or the equation does not factor neatly.

Exploring Coefficients in Quadratics

Coefficients are the numbers in front of the variables in an equation, and they help shape the curve of a quadratic equation. In the standard quadratic equation \( ax^2 + bx + c = 0 \), - \( a \) is the coefficient of the squared term, \( x^2 \), and it determines the parabola's direction.
- If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
- \( b \) is the coefficient of the linear term, and influences the parabola’s symmetry and slope.
- \( c \) is the constant term and indicates where the graph intersects the y-axis.For the equation \( 3x^2 - 9x + 2 = 0 \):
- \( a = 3 \), \( b = -9 \), and \( c = 2 \).
- Together, these coefficients define the shape and position of the parabola on the graph.
- Knowing the coefficients helps interprete and solve the equation efficiently, especially when using the quadratic formula.