Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The solutions to the quadratic equation \( -x^2 + 2x - 6 = 0 \) are \( x = 1 + \sqrt{7} \) and \( x = 1 - \sqrt{7} \).

Step 1: Rearrange the equation

First, rearrange the given equation \( -6 + 2x - x^2 = 0 \) into a standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). The equation in the standard form becomes: \( -x^2 + 2x - 6 = 0 \)

Step 2: Change the equation to the form \( (x-h)^2 = k \)

Add 6 to both sides and then divide the entire equation by -1 in order to get a positive leading coefficient: \( x^2 - 2x - 6 = 0 \) becomes \( x^2 - 2x = 6 \)

Step 3: Complete the square

To complete the square, take half of the coefficient of x, square it and add it to both sides of the equation. Here, coefficient of x is -2, half of -2 is -1, and (-1)^2 is 1. So, the equation \( x^2 - 2x = 6 \) becomes \( x^2 - 2x + 1 = 6 + 1 \), which simplifies to \( (x - 1)^2 = 7 \).

Step 4: Solve for x

Square root both sides of the equation to solve for x. So, \( x -1 = \pm \sqrt{7} \). Adding 1 to both sides to isolate x gives \( x = 1 \pm \sqrt{7} \). This simplifies to \( x = 1 + \sqrt{7} \) and \( x = 1 - \sqrt{7} \).

Step 5: Graphically verifying the solution

The graph of the given equation \( y = -x^2 + 2x - 6 \) is a parabola that opens downwards. The x-intercepts of the graph will be the solutions to the equation, since at these points, y=0. When you graph the function, you will see that the parabola indeed crosses the x-axis at these two points: \( x = 1 + \sqrt{7} \) and \( x = 1 - \sqrt{7} \), thus verifying the solution graphically.

Key Concepts

Completing the Square

In algebra, completing the square is a technique used to solve quadratic equations. It's a method that transforms a quadratic equation into a perfect square trinomial, making it easier to solve. This is a valuable tool when an equation doesn't factor nicely. Here’s how we do it:

• First, ensure your equation is in the form: \( ax^2 + bx + c = 0 \).
• Move the constant term \( c \) to the opposite side of the equation.
• Focus on the quadratic and linear terms: \( ax^2 + bx \).
• Divide every term by \( a \) if \( a eq 1 \) to make the coefficient of \( x^2 \) equal to 1.
• Take half of the coefficient of \( x \), square it, and add it to both sides to form a perfect square.

Take our example: starting with \( x^2 - 2x = 6 \), we calculate half of \(-2\), i.e., \(-1\), and square it to \(1\). Adding \(1\) to both sides gives \( x^2 - 2x + 1 = 7 \), or \( (x-1)^2 = 7 \). This equation is perfect for solving directly by taking square roots.

Graphical Verification

After solving a quadratic equation algebraically by completing the square, it's a good idea to verify your solutions graphically. This ensures you've found the correct x-values where the parabola actually crosses the x-axis. Here's a step-by-step on how graphical verification works with our equation:

• Take the quadratic function, e.g., \( y = -x^2 + 2x - 6 \).
• Graph the function using a graphing tool or calculator.
• Check where the graph intersects the x-axis; these intersection points are your solutions.

In our problem, we know the solutions derived from completing the square are \( x = 1 + \sqrt{7} \) and \( x = 1 - \sqrt{7} \). Graphing verifies that the parabola intersects the x-axis precisely at these points, confirming our solution is correct.

Parabolas

Parabolas are the graphs of quadratic functions, shaped like a U or an upside-down U. They have unique properties, such as a vertex, axis of symmetry, and potential x-intercepts. Understanding these features helps when working with quadratic equations.

• For a quadratic equation \( ax^2 + bx + c = 0 \), the graph will be a parabola.
• If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards.
• The vertex is the highest or lowest point, acting as a crucial point for transformations.
• The axis of symmetry is a vertical line through the vertex dividing the parabola into mirror images.
• X-intercepts are where the graph crosses the x-axis, representing the solutions of the equation.

In our example, the parabola from \( y = -x^2 + 2x - 6 \) opens downwards because the leading coefficient \(-1\) is negative, with the vertex and axis of symmetry critical in pinpointing the graph’s behavior.

Solutions to Quadratic Equations

Quadratic equations can have either real number solutions or complex solutions. The solutions are the x-values where the quadratic touches or crosses the x-axis, known as zeros or roots.

• There are several ways to find these solutions: factoring, using the quadratic formula, or completing the square.
• Real solutions occur where the graph intersects the x-axis.
• If the graph doesn’t cross the x-axis, the equation's solutions are complex (involving imaginary numbers).
• The number of solutions depends on the discriminant \( b^2 - 4ac \) from the quadratic formula.

In our quadratic equation \( -6 + 2x - x^2 = 0 \), completing the square revealed two real solutions \( x = 1 + \sqrt{7} \) and \( x = 1 - \sqrt{7} \). The graphical verification confirmed these are points where the parabola intersects the x-axis.