Problem 59 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ x^{2}+10 x+28=0 $$

Step-by-Step Solution

Answer

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

Step 1: Move Constant to Right Side

Begin by moving the constant term to the right side of the equation. Start with the equation: \[ x^{2}+10x+28=0 \]Subtract 28 from both sides to get:\[ x^2 + 10x = -28 \]

Step 2: Find the Value to Complete the Square

Find the number needed to complete the square, by taking half of the coefficient of x, then squaring it. Here, the coefficient of x is 10.Half of 10 is 5, and squaring it results in 25. Add this value to both sides of the equation to maintain the equality.\[ x^2 + 10x + 25 = -28 + 25 \]

Step 3: Write as a Perfect Square

The left side of the equation is now a perfect square trinomial, which can be written as:\[ (x+5)^2 = -3 \]

Step 4: Solve the Perfect Square Equation

Take the square root of both sides to solve for x:\[ x + 5 = \pm \sqrt{-3} \]Since the square root of a negative number involves imaginary numbers, write:\[ x + 5 = \pm i\sqrt{3} \]

Step 5: Solve for x

Isolate x by subtracting 5 from both sides:\[ x = -5 \pm i\sqrt{3} \]

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental expressions in algebra, characterized by the presence of an \( x^2 \) term. Essentially, any equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \), is a quadratic equation.The purpose of solving quadratic equations is typically to find the value(s) of \( x \) that satisfy the equation. These values are often called the roots or solutions of the equation. Solving quadratics is crucial because they appear not only in mathematical problems but also in real-life scenarios, like in physics, engineering, and budgeting problems.In this exercise, we used a method known as "completing the square" to find the solutions. This method manipulates the initial equation into a format that simplifies finding the roots. It's a vital technique for understanding more complex algebraic concepts, and it sets the stage for grasping how functions behave graphically.

Finding Perfect Square Trinomials

A perfect square trinomial is a unique form of expression that can be expressed as the square of a binomial. These have the general form \( (x + m)^2 \), which expands to \( x^2 + 2mx + m^2 \).In the process of completing the square, we aim to transform part of our quadratic equation into this form. Here's how it worked in the given problem:

• First, observe the equation \( x^2 + 10x = -28 \).
• Take the coefficient of \( x \), which is 10, halve it to get 5, and then square it to get 25.
• By adding 25 to both sides, we obtain a perfect square trinomial on the left side: \( x^2 + 10x + 25 \).
• This trinomial is equivalent to \( (x + 5)^2 \), meaning we successfully transformed the quadratic into the square of a binomial.

Understanding how to form and recognize a perfect square trinomial is essential for anyone aiming to master the technique of completing the square. This step bridges the manipulation of equations with the geometric concept of squares, increasing our algebra toolkit.

Exploring Imaginary Numbers

Imaginary numbers extend our number system to solve problems that real numbers alone cannot address. The basic unit of imaginary numbers is \( i \), defined as \( i = \sqrt{-1} \). This implies that \( i^2 = -1 \).In the solution of the quadratic equation \((x+5)^2 = -3\), taking the square root of both sides reveals an imaginary component:

• We found \( x+5 = \pm \sqrt{-3} \).
• A negative value under a square root necessitates the use of \( i \): \( \pm i\sqrt{3} \).

Thus, the solutions \( x = -5 \pm i\sqrt{3} \) show that the equation has complex roots, which includes both a real component (-5) and an imaginary component (\( \pm i\sqrt{3} \)).Imaginary numbers, together with real numbers, make up complex numbers. This concept is foundational in advanced mathematics, with applications in fields such as engineering and computer science, where phenomena are described not just by magnitude, but also by direction.