Problem 86 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(9-6 x+x^{2}=0\)

Step-by-Step Solution

Answer

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

Step 1: Identify the coefficients

Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation \(9-6 x+x^{2}=0\). To make it simply clearer, we can rewrite that as \(x^{2} - 6x + 9 = 0\), where \(a=1\), \(b=-6\) and \(c=9\).

Step 2: Apply the quadratic formula

Apply the quadratic formula \(-b ± √(b^{2} - 4ac) / (2a)\) to find the roots of the equation.

Step 3: Substitute the given values

Substitute \(a=1\), \(b=-6\) and \(c=9\)and calculate the values of \(x\). This will give: \(x = (6 ± √((-6)^{2} - 4*1*9 )) / (2*1)\)

Step 4: Evaluate the values

Evaluate the values in the square root and then the entire expression. It comes out to be 6 ± √((36 - 36)) / 2, which simplifies to \(x = 6/2 = 3\).

Step 5: Checking the roots

Substitute \(x = 3\) back into the original equation to verify. \(9 - 6 * 3 + 3^{2} = 9 - 18 + 9 = 0\)

Step 6: Solution

Since \(9 - 6 * 3 + 3^{2} = 0\) is a true statement, it verifies that \(x = 3\) is indeed a root of the equation. Since the discriminant (b² - 4ac) equals zero, the quadratic equation has exactly one real and repeated root

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations. It is derived from the process of completing the square and is represented by the formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the solutions (roots) to any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is especially useful when factoring is difficult or when the roots are not obvious integers or fractions.
Here's how it works:

• The \(b\) in the formula represents the coefficient of the linear term (\(x\)).
  
• \(a\) is the coefficient of the quadratic term \(x^2\), and \(c\) is the constant term.
  
• The expression under the square root sign, \(b^2 - 4ac\), is called the discriminant and plays a significant role in determining the nature of the roots.

Using this formula, you can find the values of \(x\) that satisfy the equation, efficiently and reliably.

Roots of Equations

The roots, or solutions, of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Understanding the nature of these roots can help us graph the equation and understand the parabola it represents.
Depending on the discriminant (\(b^2 - 4ac\)), an equation can have different types of roots:

• If the discriminant is greater than zero, the equation has two distinct real roots.
  
• If the discriminant equals zero, the equation has one real root, or more specifically, a repeated or "double" root.
  
• If the discriminant is less than zero, the equation has two complex roots, which are not real numbers.

For example, in our quadratic equation, after calculation, the discriminant was found to be zero. This ensures that our equation has exactly one real and repeated root, simplifying the solving process.

Discriminant

The discriminant is a critical part of the quadratic formula, written as \(b^2 - 4ac\). It gives insightful information about the nature of the roots of a quadratic equation. By calculating the discriminant, you can predict:

• Two distinct real roots, if the discriminant is positive.
  
• One repeated real root, if the discriminant is zero. This indicates the parabola touches the x-axis at one point.
  
• Two complex roots, both non-real, if the discriminant is negative. This means the parabola does not intersect the x-axis at all.

In the example equation \(x^2 - 6x + 9 = 0\), the discriminant was computed to be zero, confirming a single, repeated root, which significantly simplifies our problem-solving process.

Solving Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true, meaning both sides of the equation are equal. There are several methods to solve quadratic equations:

• The first approach is factoring, which involves finding common factors shared between terms.
  
• Another highly reliable method is using the quadratic formula, particularly useful when equations are not easily factorable.
  
• Completing the square is another process that can be used, though it might involve more steps than the quadratic formula or factoring.
  
• Finally, graphing the equation on a coordinate plane and identifying where it intersects the x-axis can provide visual insight into the solutions.

In our example, \(9-6x+x^2=0\), rewriting it as \(x^2 - 6x + 9 = 0\) and using the quadratic formula clearly showed the root is \(x = 3\), a result confirmed by substituting \(x = 3\) back into the original equation.