Problem 51 - College Algebra Essentials

Question

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}-6 x-11=0 $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the quadratic equation

First, rewrite the given quadratic equation \(x^{2}-6 x-11=0\) in the form \(x^{2}-6 x=11\). This is done by moving the constant term to the right side of the equation.

Step 2: Complete the square on the left side

Next, 'complete the square' on the left side of the equation. This means finding a number which makes the expression \(x^2 - 6x\) a perfect square. This is done by taking half the coefficient of x, squaring it, and adding it to both sides of the equation. The coefficient of x is -6, its half is -3, and (-3)^2 equals 9. Hence, the equation becomes \((x^{2}-6 x+9)=11+9\)

Step 3: Simplify the equation

Simplify both sides of the equation to get \((x-3)^2 = 20\)

Step 4: Solve for x

Finally, solve for x by taking the square root of both sides of the equation and considering both positive and negative roots, to get \(x=3±\sqrt{20}\), which simplifies to \(x=3±2\sqrt{5}\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that are set to zero and involve a variable raised to the second power, generally expressed in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). Understanding how to solve these equations is a fundamental skill in algebra.

One of the methods to solve quadratic equations is by completing the square which requires transforming the equation into a perfect square trinomial. By doing so, students can find solutions by manipulating the equation into a format that allows taking square roots easily.

Perfect Square Trinomials

Perfect square trinomials are quadratic expressions that can be written as the square of a binomial. They take the form \( (x + p)^2 = x^2 + 2px + p^2 \) where \( p \) is a real number. The process of completing the square involves modifying a given quadratic equation such that one side becomes a perfect square trinomial.

For instance, in the equation from the exercise, \( x^2 - 6x \) is transformed into \( (x - 3)^2 \) by adding \(9\), the square of half of \(6\), to both sides. Recognizing and creating perfect square trinomials allow for simplification using the property that \( (x + p)(x + p) = (x + p)^2 \).

Quadratic Formula

The quadratic formula is a powerful algebraic solution that can be used to find the roots of any quadratic equation. It is derived from the process of completing the square and is stated as \( x = \frac{{-b \(u00B1\) \sqrt{b^2 - 4ac}}}{2a} \). This formula provides the exact solutions by calculating the roots independently of factoring or completing the square, which can be particularly useful for equations where factoring is not easily apparent.

Using the quadratic formula can be a safe fallback method if other approaches fail or are cumbersome, as it applies universally to all quadratic equations.

Algebraic Methods

Algebraic methods encompass a range of techniques used to manipulate and solve equations, including completing the square and the quadratic formula. These methods rely on algebraic rules and properties such as the distributive law, combining like terms, and the properties of equality.

Completing the square is specifically beneficial as it not only helps to solve quadratic equations but also aids in understanding the graphical representation of quadratics, such as parabolas, and in the derivation of other methods like the quadratic formula. Mastery of these algebraic methods is essential in building a strong foundation for further study in mathematics.