Problem 50 - College Algebra Essentials

Question

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+4 x=12 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2\) and \(x = -6\).

Step 1: Rearrange the equation

Add -12 on both sides of the equation to keep only terms with x on the one side, which gives the equation \(x^{2} + 4x - 12 = 0\)

Step 2: Complete the square

The 'completing the square' strategy involves rearranging the equation to make it a perfect square on one side. To do this, half the coefficient of x, square it and then add it to both sides. Here, halving the coefficient of x gives 2, and squaring gives 4. The equation then becomes \(x^{2} + 4x + 4 = 12 + 4\) which simplifies to \((x + 2)^{2} = 16\).

Step 3: Solve for x

To find the value of x, take the square root of both sides, remembering that there are two solutions: a positive and a negative. Therefore, \(x + 2 = \pm 4\). Finally, subtract 2 from both sides of the equation to solve for x, which gives \(x = \pm4 - 2\).

Key Concepts

Quadratic Equations

Quadratic equations form the foundation of many mathematical concepts in algebra. They are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The highest degree of the polynomial is 2, which is why they are called "quadratic." These equations are essential in various fields, from solving physics problems to modeling real-world scenarios like projectile motion.

Here's the basic structure:

• ***\(x^2\) term***: The quadratic term, which is always present.
• ***\(x\) term***: The linear term, which might sometimes be zero.
• ***Constant term***: A number without variables.

To solve quadratic equations, there are several methods available, such as factoring, using the quadratic formula, or completing the square. Completing the square is particularly useful when the equation does not easily factorize.
Understanding how to manipulate these equations is crucial for progressing in mathematics and applying it to complex problems.

Perfect Square Trinomial

The concept of a perfect square trinomial is instrumental in completing the square. It is an expression that can be factored into a square of a binomial. For example, \((x + a)^2 = x^2 + 2ax + a^2\).

To transform a quadratic equation into a perfect square trinomial, follow these:

• ***Half the coefficient of x:*** Look at the term multiplying \(x\). Divide it by 2, as seen in the equation \(x^2 + 4x = 12\), where the coefficient of \(x\) is 4. Half of 4 is 2.
• ***Square that result:*** Now, square that halved number to get 4.
• ***Add to both sides:*** By adding this square to both sides, you form a trinomial that can be factored as a binomial squared. Our example becomes \((x + 2)^2 = 16\).

Creating a perfect square trinomial simplifies solving by making it easy to apply square roots and find solutions to the equation.

Solving Equations

The final step in completing the square is solving the equation after transforming it into a perfect square trinomial. Following the example \((x + 2)^2 = 16\), here's the basic process:

• ***Take the square root of both sides:*** Here, you would compute \(\sqrt{(x + 2)^2}\) and \(\sqrt{16}\). This yields \(x + 2 = \pm 4\). Always remember that taking the square root can produce two potential solutions, positive and negative.
• ***Solve for x:*** Isolate \(x\) by subtracting 2 from each side. This results in two equations: \(x = 4 - 2\) and \(x = -4 - 2\), simplifying to \(x = 2\) and \(x = -6\).

The method of completing the square yields precise solutions by reducing a quadratic equation into two linear equations. It highlights the symmetry in quadratic equations and provides deeper insight into their solutions, making it an invaluable skill in algebra.