Problem 11 - Algebra 2

Question

Solve each equation by factoring. Then graph. \(x^{2}+9=6 x\)

Step-by-Step Solution

Answer

The solution is \(x = 3\), with a parabola touching the x-axis at this point.

Step 1: Rewrite the Equation

Start by rewriting the given equation. We need to set one side to zero for easier factoring:\[x^2 + 9 = 6x \]Subtract \(6x\) from both sides to achieve this:\[x^2 - 6x + 9 = 0\]

Step 2: Factor the Quadratic Equation

We have: \[x^2 - 6x + 9 = 0\]To factor this, look for two numbers that multiply to \(9\) (the constant term) and add to \(-6\) (the coefficient of \(x\)). These numbers are \(-3\) and \(-3\). Thus, we can factor the expression as:\[(x - 3)(x - 3) = 0\]This can be simplified further to:\[(x - 3)^2 = 0\]

Step 3: Solve for x

Since the factored form of the equation is \((x - 3)^2 = 0\), we set the expression inside the parenthesis to zero:\[x - 3 = 0\]Add \(3\) to both sides to solve for \(x\):\[x = 3\]So, the solution is \(x = 3\).

Step 4: Graph the Solution

Since \((x - 3)^2 = 0\) represents a perfect square, it indicates a repeated root. The graph of the quadratic \(x^2 - 6x + 9\) is a parabola opening upwards with its vertex touching the x-axis at the point \((3, 0)\). Plot the graph with the vertex at \(x = 3\) and note that it does not cross the x-axis at any other point.

Key Concepts

Graphing Parabolas

When graphing a parabola, especially when given a quadratic equation like \(x^2 - 6x + 9 = 0\), it's important to understand its geometric properties. A parabola is a U-shaped curve that can either open upwards or downwards. The general equation of a quadratic function, \(ax^2 + bx + c = 0\), helps determine the direction of the parabola:

• If \(a > 0\), the parabola opens upwards, resembling a \"smiling face\".
• If \(a < 0\), it opens downwards, resembling a \"frown.\"

For the equation \(x^2 - 6x + 9\), the coefficient of \(x^2\) is positive (\(a = 1\)), indicating the parabola opens upwards. The vertex form of a parabola, derived from factoring the equation, can also provide the vertex of the curve. Here, the vertex is \((3, 0)\), meaning the parabola touches the x-axis at the point \((3, 0)\) and does not cross it again. This property is crucial for sketching the graph because it tells us that the solution \(x = 3\) is a repeated root.
Bringing it all together, plot your x-values and corresponding \(y\)-values using your factored equation, and draw the smooth, continuous curve of the parabola, ensuring it opens upwards and highlights the vertex as a key feature.

Solving Polynomial Equations

Polynomial equations, such as the quadratic equation in this example, involve expressions with more than one term. Solving them often requires factoring, which simplifies the equation into products of simpler binomials or polynomials. Here are some tips when tackling these:

• First, arrange the polynomial so that all terms are on one side of the equation set to zero. This step facilitates factoring, as seen when transforming \(x^2 + 9 = 6x\) into \(x^2 - 6x + 9 = 0\).
• Next, identify factor pairs that multiply to the constant term and add to the linear coefficient. These steps narrow down potential solutions and lead to a factored form, like \((x - 3)^2\).
• Lastly, solve for \(x\) by setting each factor equal to zero and finding the x-values.

Solving polynomial equations involves finding the x-intercepts or roots, which are critical solutions where the graph intersects the x-axis. This is particularly highlighted in this problem with the solution \(x = 3\), where the parabola's vertex also serves as its sole intercept with the x-axis.

Quadratic Functions

Quadratic functions are a specific type of polynomial function where the highest degree term is squared. They are generally expressed in the form \(f(x) = ax^2 + bx + c\). Unlike linear functions, where the graph is a straight line, quadratic functions produce a parabola shape. Here's what you need to know about them:

• The vertex form, \(f(x) = a(x - h)^2 + k\), shows the vertex \((h, k)\) and provides insight into the graph's behavior.
• The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), also offers a direct method to find the roots of the equation but wasn't needed in this example due to the simplicity of factoring.
• And an essential aspect of quadratic functions is their symmetry concerning the vertex, reflecting equal distance to either side of it for identical y-values.

In our exercise, we dealt with \(x^2 - 6x + 9\), a perfect square trinomial factored as \((x - 3)^2\). This presented just one unique solution, a defining feature of this quadratic equation's graph. Understanding these elements allows a deeper grasp into how quadratic functions operate and their implications in both graphing and solving contexts.