Problem 48 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$2 x^{2}-3 x+1=0$$

Step-by-Step Solution

Answer

The two solutions for the given quadratic equation are \(x = 1\) and \(x = 0.5\).

Step 1 - Identify the coefficient values

Firstly, identify the coefficients \(a\), \(b\), and \(c\) in the given quadratic equation \(2x^2 - 3x + 1 = 0\), where \(a = 2\), \(b = -3\) and \(c = 1\). These coefficients will be substituted into the quadratic formula.

Step 2 - Substitute coefficients into the quadratic formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the quadratic formula. To solve for \(x\), this would be \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4*2*1}}{2*2}\).

Step 3 - Simplify the equation

Simplify under the square root first. This is \( (-3)^2 - 4*2*1 = 9 - 8 = 1 \). The equation becomes \(x = \frac{3 \pm \sqrt{1}}{4}\).

Step 4 - Compute the final solutions

Lastly, calculate the two possible solutions for \(x\), one using the plus sign, and one using the minus sign: \(x = \frac{3 + 1}{4} = 1\) and \(x = \frac{3 - 1}{4} = 0.5\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are fundamental in algebra, and they come in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known values with \( a \) not equal to zero. To solve these equations, one might use various methods such as factoring, completing the square, or using the quadratic formula.

The quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a reliable method that always works and is especially useful when the equation is not easily factorable. For the equation \( 2x^2 - 3x + 1 = 0 \), we applied the quadratic formula.

Understanding the Quadratic Formula:

Remember that the symbol \( \pm \) means there will be two solutions, one with a plus and one with a minus. After substituting values and simplifying, we found the solutions to be \( x = 1 \) and \( x = 0.5 \).

When solving, it's crucial to follow each step methodically to avoid errors, especially while simplifying the square root and rationalizing the denominator. These steps lead to the precise answers that can be verified by substituting back into the original equation.

Coefficients in Quadratics

Understanding the role of coefficients in a quadratic equation is essential. In the equation \( ax^2 + bx + c = 0 \), \( a \) is the coefficient of the squared term, \( b \) is the coefficient of the linear term, and \( c \) is the constant term.

Each of these coefficients influences the equation's graph and solutions in different ways:

• The coefficient \( a \) determines the parabola's direction of opening and its width.
• The coefficient \( b \) affects the position of the parabola along the x-axis.
• The coefficient \( c \) represents the y-intercept, the point where the graph crosses the y-axis.

The correct identification of coefficients is crucial when applying the quadratic formula. For the equation given, \( 2x^2 - 3x + 1 = 0 \), we have \( a = 2 \), \( b = -3 \), and \( c = 1 \). Mistaking these values can lead to incorrect solutions. Always ensure to maintain the signs of the coefficients as they indicate the direction of the operation.

Simplifying Radicals

The process of simplifying radicals involves reducing the expression under the square root to its simplest form. When solving quadratic equations with the quadratic formula, the square root part, \( \sqrt{b^2 - 4ac} \), often requires simplification.

In our example, we simplified \( \sqrt{(-3)^2 - 4*2*1} = \sqrt{1} \), which simplifies to 1. When the value under the square root is a perfect square, simplifying is straightforward as we just take the square root of that number. However, when it isn't a perfect square, simplification involves factoring out perfect squares and leaving the remaining factors inside the radical.

Essential Tips for Simplification:

• Break down the number into its prime factors.
• Pair up the prime factors and move them out of the radical.
• Simplify any fractions under the square root separately before or after taking the square root, as needed.

Accuracy in simplifying radicals prevents errors in the final answer and also ensures that the solutions are presented in their simplest form, which is important in math and required in many application areas.