Problem 15 - Algebra and Trigonometry

Question

Solve the equation by completing the square. \(x^{2}-6 x-11=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 3 + 2\sqrt{5}\) and \(x = 3 - 2\sqrt{5}\).

Step 1: Move Constant to the Right Side

Start by isolating the quadratic and linear terms. Move the constant term to the right side of the equation:\[x^2 - 6x = 11\]

Step 2: Complete the Square

To complete the square, find the number that will turn the left side into a perfect square trinomial. Take half the coefficient of \(x\), square it, and add it to both sides.Half of \(-6\) is \(-3\). Squaring it gives \(9\). Add \(9\) to both sides:\[x^2 - 6x + 9 = 11 + 9\]This becomes:\[(x - 3)^2 = 20\]

Step 3: Solve for \(x\)

Solve for \(x\) by taking the square root of both sides of the equation:\[x - 3 = \pm \sqrt{20}\]\[x - 3 = \pm 2\sqrt{5}\]Add 3 to both sides to solve for \(x\):\[x = 3 \pm 2\sqrt{5}\]

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. It transforms a standard quadratic equation, like \( ax^2 + bx + c = 0 \), into a perfect square trinomial form. Here’s how it works:

• Move the constant term to the other side of the equation. This isolates the quadratic and linear terms.
• Take half of the coefficient of the linear term (\(b\)), square it, and add this square to both sides. This process helps form a perfect square trinomial on the left side.
• Rewrite the trinomial as the square of a binomial. You can now solve it by taking the square root of both sides to find the values of \(x\).

By mastering completing the square, you can solve any quadratic equation, even when it doesn't factor easily. Much like in our problem where \( x^2 - 6x - 11 = 0 \), we transformed it into \((x - 3)^2 = 20\) to find our solutions.

Quadratic Equation

A quadratic equation is a second-degree polynomial, typically written in the form \(ax^2 + bx + c = 0\). It represents a parabola on a graph, and its solutions correspond to the points where the parabola intersects the x-axis.

Key features of quadratic equations include:

• The leading coefficient \(a\) impacts the parabola's direction. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.
• The solutions can be found using methods like factoring, completing the square, or the quadratic formula.
• Quadratic equations can have two solutions, one solution, or no real solutions, determined by the discriminant \(b^2 - 4ac\).

In the example \(x^2 - 6x - 11 = 0\), we used completing the square to transform and solve for the points where the equation equaled zero.

Algebraic Techniques

Algebraic techniques are the methods used to solve equations and simplify expressions. For quadratic equations, several techniques can be applied, such as:

• Factoring: Breaking down the quadratic into multiplicative components, useful when the equation is easily factorable.
• Completing the Square: Turning the quadratic into a perfect square trinomial, as we did by adding \(9\) to both sides of the equation.
• Quadratic Formula: A universal solution method, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), useful when other methods are cumbersome.

These techniques provide flexibility in solving varied quadratic problems. Understanding when and how to apply each technique is essential, making algebra a powerful tool for solving complex problems efficiently. Our example highlights completing the square, showcasing how an algebraic method simplifies solving a quadratic equation.