Problem 36 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possib $$9-6 x+x^{2}=0$$

Step-by-Step Solution

Answer

The solution for the equation \(x^{2} - 6x + 9 = 0\) is \(x = 3\).

Step 1: Identify the coefficients

In the given equation \(x^{2} - 6x + 9 = 0\), identify the coefficients \(a\), \(b\), and \(c\). In this case, \(a = 1\), \(b = -6\), and \(c = 9\).

Step 2: Substitute the coefficients in the quadratic formula

Now substitute these values into the quadratic formula \(x = \[-b \pm sqrt(b^{2} - 4ac)\]/(2a)\). Therefore we get \(x = \[6 \pm sqrt((-6)^{2} - 4*1*9)\]/(2*1)\).

Step 3: Execute the operations

Perform the operations inside the square root first, then execute remaining operations. \(x = \[6 \pm sqrt(36 - 36)\]/2 = \[6 \pm sqrt(0) \]/ 2 = 3\).

Key Concepts

Solving Equations

Solving quadratic equations is a fundamental skill in algebra. A quadratic equation is a polynomial equation of the second degree. This means that the highest power of the variable, often represented as \( x \), is squared. The general form of a quadratic equation is \( ax^2 + bx + c = 0 \).
To solve such equations, you can use various methods including factoring, completing the square, or the quadratic formula. Each method has its own advantages depending on the complexity of the equation.

• Factoring is useful when the quadratic is easily decomposable into two binomials.
• Completing the square is beneficial for converting the equation into a perfect square trinomial.
• The quadratic formula is universally applicable and provides solutions for all kinds of quadratic equations, regardless of whether they can be factored easily.

The goal is always to find the values of \( x \) that satisfy the equation, which are also known as the roots of the equation.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It provides a standardized way to find the roots of the equation \( ax^2 + bx + c = 0 \). The formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here’s how it works:

• First, identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Plug these values into the formula.
• Calculate the discriminant, which is the part \( b^2 - 4ac \). This will determine the nature of the roots.
• Solve the equation by completing the indicated operations - subtraction, addition, and division.

The quadratic formula is especially useful because it can find solutions even when the quadratic cannot be factored or when the solutions are irrational.
It is important to pay attention to the discriminant because it tells you if the solutions are real, repeated, or complex. When the discriminant is zero, the equation has exactly one real solution.
Otherwise, it may yield two solutions.

Irrational Solutions

Not all solutions to quadratic equations are nice, neat whole numbers. Sometimes, you'll encounter irrational solutions. These solutions involve square roots of non-perfect squares, leading to non-repeating, non-terminating decimals. Here’s what to know:

• When using the quadratic formula, if the discriminant \( b^2 - 4ac \) is not a perfect square, the roots are irrational.
• Irrational solutions are often written in terms of square roots for precision rather than converted entirely into decimals.
• In many mathematical and real-world applications, it's crucial to simplify these solutions by removing any factors from the square root when possible.

For instance, in the given example, the discriminant was zero, leading to a rational solution. However, if the discriminant were a number like 5 or 7, the solution would be irrational.
Learning to work with irrational numbers enhances your ability to handle more complex mathematical problems and ensures a comprehensive understanding of algebraic concepts.