Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$x^{2}-6 x+2=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation are \(x=3+\sqrt{7}\) and \(x=3-\sqrt{7}\).

Step 1: Write the quadratic equation

The given quadratic equation is \(x^{2}-6x+2=0\).

Step 2: Rearrange the equation

We can rewrite the equation as \(x^{2}-6x=-2\). The goal is to form a perfect square trinomial on the left side.

Step 3: Complete the square

To complete the square on the left side, the coefficient of x should be halved and squared which gives \((-6/2)^2=9\). We add this to both sides of the equation. Then our equation becomes \(x^{2}-6x+9=-2+9\), which simplifies to \((x-3)^2=7\).

Step 4: Solve for x

To solve for x, take the square root of each side. Don't forget to consider both the positive and negative root. We then have \(x-3=\sqrt{7}\) and \(x-3=-\sqrt{7}\). Adding 3 to each of these gives us the roots \(x=3+\sqrt{7}\) and \(x=3-\sqrt{7}\).

Step 5: Verify graphically

To verify the solution graphically, plot the function \( y = x^{2}-6x+2\). The roots found should be the x-intercepts of the graph. If the graph of the function crosses the x-axis at \(x=3+\sqrt{7}\) and \(x=3-\sqrt{7}\), this confirms the solutions are correct.

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental tool in algebra. It has the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This equation represents a parabola when graphed on a coordinate plane.

The equation can have:

• Two distinct real roots
• One real double root
• No real roots (complex or imaginary roots)

The quadratic equation given, \( x^2 - 6x + 2 = 0 \), is an example of how quadratics can appear in various problems, whether in geometry, physics, or everyday scenarios.

Understanding the structure of quadratic equations is essential for solving them efficiently. They pop up to describe everything from product cost functions to modeling projectile motion.

Perfect Square Trinomial

Creating a perfect square trinomial is a critical step in the method of completing the square. A perfect square trinomial is an expression that can be written in the form \((x + h)^2\).

To create a perfect square trinomial from the quadratic equation \( x^2 - 6x + 2 = 0 \), we focus on the \( x^2 - 6x \) part.

Here’s how we transform it:

• First, identify the coefficient of \(-6\) on \(x\).
• Halve it to get \(-3\), and then square this result to obtain \(9\).
• Add and subtract this \(9\) on the left-hand side to maintain balance.

This turns the expression into \((x - 3)^2\), making it a perfect square trinomial. With the constant on the other side, the equation becomes \((x - 3)^2 = 7\).

This transformation makes solving for \(x\) straightforward by using basic algebraic steps.

Solution of Quadratic Equations

Solving quadratic equations, such as \((x - 3)^2 = 7\), often involves finding the roots that make the equation true. Here’s the process broken down:

First, take the square root of both sides of the equation

• \((x - 3)^2 = 7\)
• Taking the square root gives \(x - 3 = \pm \sqrt{7}\)

Doing so, we must consider both the positive and negative roots due to the nature of squaring.

Next, solve for \(x\):

• For \(x - 3 = \sqrt{7}\), add 3 to both sides to find \(x = 3 + \sqrt{7}\).
• For \(x - 3 = -\sqrt{7}\), likewise, add 3 to get \(x = 3 - \sqrt{7}\).

These solutions \(x = 3 \pm \sqrt{7}\) are the x-intercepts of the function, representing where the parabola crosses the x-axis on the graph.

Graphical Verification

To confirm the solutions found algebraically, graphical verification is a reliable method. This process involves plotting the graph of the quadratic function. Here, we consider the function \( y = x^2 - 6x + 2 \).

When you draw this parabola, its roots, which were calculated as \(x = 3 + \sqrt{7}\) and \(x = 3 - \sqrt{7}\), should appear as the points where the graph intersects the x-axis.

• These x-intercepts indicate the real roots of the equation.
• The vertex of this graph is located at \(x = 3\), along the line of symmetry, confirming the balance of our completed square.

Graphically solving a quadratic provides an excellent visual check that complements algebraic techniques. It's especially helpful for visual learners to see the connection between the equation and its graphical representation.