Problem 19 - 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 Other Side

Start by moving the constant, -11, to the other side of the equation. The equation becomes:\[x^2 - 6x = 11\]

Step 2: Identify the Coefficient of x-term

Identify the coefficient of the \(x\)-term, which is -6. This will be used to find the number to complete the square.

Step 3: Calculate the Number to Complete the Square

Take the coefficient from the \(x\)-term, divide it by 2, and then square it.\[\left(\frac{-6}{2}\right)^2 = 9\]We will add and subtract this value in the equation to complete the square.

Step 4: Add and Subtract Calculated Number

Add and subtract the number calculated in Step 3 (9) inside the equation to maintain equality:\[x^2 - 6x + 9 - 9 = 11\]

Step 5: Rewrite the Equation as a Perfect Square

Rewrite the expression \(x^2 - 6x + 9\) as a square of a binomial:\[(x - 3)^2 = 11 + 9\]Simplify to get:\[(x - 3)^2 = 20\]

Step 6: Solve for x

Take the square root of both sides:\[x - 3 = \pm \sqrt{20}\]Simplify \(\sqrt{20}\) to \(2\sqrt{5}\), giving:\[x - 3 = \pm 2\sqrt{5}\]

Step 7: Isolate x

Add 3 to both sides to solve for \(x\):\[x = 3 \pm 2\sqrt{5}\]

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest exponent of the variable is 2. This means they are always of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a eq 0\). When you graph a quadratic equation, you get a parabola, which can open upwards or downwards depending on the sign of \(a\).

Quadratics can appear in many different real-world situations, like calculating areas, forming curves in design, and predicting statistics. Understanding how to solve them is crucial, especially when using different methods like factoring, using the quadratic formula, or completing the square to find the roots. In the problem we solved, completing the square was used, but each method hinges on bringing the equation to an understandable state.

Solving Equations

Solving an equation involves finding the value(s) of the variable that make the equation true. For quadratic equations such as \(x^2 - 6x - 11 = 0\), we want to determine the values of \(x\) that will satisfy this equation. There are several techniques available for this:

• Factoring: This works when a quadratic can be expressed as a product of two binomials.
• Quadratic Formula: Derived from the standard form, helpful when factoring is complex. The formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\).
• Completing the Square: This technique involves creating a perfect square trinomial from the quadratic equation, allowing for easier solving.

In our example, completing the square led us to \((x - 3)^2 = 20\), making it straightforward to find the solutions through taking square roots and further simplifying.

Algebraic Manipulation

Algebraic manipulation is a fundamental skill that involves changing the form of an expression or equation without changing its value. This process is vital in solving equations because it allows us to work with equations in a way that reveals solutions more easily.

To achieve this effectively, you often perform the following operations:

• Rearranging Terms: Moving terms across the equality to simplify or isolate variables, as seen when we moved \(-11\) to the right side of the equation.
• Combining Like Terms: Grouping similar terms on each side to keep the equation tidy, though not explicitly shown here, is often used in algebra.
• Adding/Subtracting Values: Adding and subtracting the same number (as we did with 9 here) to both sides of an equation to maintain balance.
• Factoring and Expanding: Breaking down or expanding expressions, like rewriting \((x-3)^2\) from \(x^2 - 6x + 9\).

Such manipulations help in transforming quadratic equations into simpler forms for easy solution finding, ensuring the equation retains its balance throughout the process.