Problem 96 - College Algebra

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$9-6 x+x^{2}=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(9 - 6x + x^2 = 0\) is \(x = 3\). As factoring gives a squared term, it is a repeated root.

Step 1: Rearranging the Equation

First, rearrange the quadratic equation given in normal form, which is \(ax^2 + bx + c = 0\). In this case, the given equation is rearranged to \(x^2 - 6x + 9 = 0\).

Step 2: Selecting a Method to Solve the Equation

Choose a method to solve the quadratic equation. Since the equation's coefficients are whole numbers, the factoring method steps could be followed.

Step 3: Solve the Equation by Factoring

Therefore, factoring the quadratic equation gives us \((x - 3)^2 = 0\). The solutions to the equation are found by setting \(x - 3 = 0\).

Step 4: Find the Value of x

Finally, solving \(x - 3 = 0\) gives us a single solution \(x = 3\). As the expression is squared, it is a repeated root.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a common method used to solve them, especially when dealing with integers as coefficients. The basic idea is to break the quadratic expression into simpler linear factors from which the solution can be determined. For the equation \[x^2 - 6x + 9 = 0\] we can see if it can be expressed as \[(x - a)(x - b) = 0\]. In this example, you'll notice the expression can be rearranged to\[(x - 3)^2 = 0,\] which suggests that both roots are the same.The benefits of factoring include:

• Simplicity: When numbers fit nicely, this is usually the fastest method.
• Insight into the roots: Factoring helps to easily identify the roots of the equation directly.

Always check your factors by expanding them back. If they reconstruct the original quadratic equation, you factored correctly.

Quadratic Equation

A quadratic equation is any equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) represents the variable to be solved for. The characteristic 'U' shape curve they graph is known as a parabola.The key features of quadratic equations include:

• They have degree 2, meaning the highest power of \(x\) is 2.
• They can have different numbers of solutions (real or complex), up to two distinct solutions, or one repeated root.

Quadratics can be solved using various methods including factoring, completing the square, or using the quadratic formula:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]In our example, \(x^2 - 6x + 9 = 0\), the equation naturally factored nicely, showing its roots smoothly through the factorization.

Repeated Root in Quadratics

When solving quadratic equations, sometimes, you will encounter a repeated root, also known as a double root. This happens when the quadratic can be expressed in the form \((x - a)^2 = 0\), indicating the value of \(x\) that satisfies the equation is the same. In such scenarios, the solution is straightforward as there is only one unique answer for \(x\). For example, in the equation\(x^2 - 6x + 9 = 0\), the factorization was \((x - 3)^2\), showing that\(x = 3\) is a repeated root.Situations with repeated roots imply:

• The parabola touches the x-axis at a single point.
• The discriminant \(b^2 - 4ac\) is zero \((0)\), indicating that the quadratic has a perfect square form.

This concept is crucial since it changes the graphical interpretation of the solution, emphasizing the nature of its roots and symmetrical properties of the quadratic function.