Problem 41 - Algebra 2

Question

Solve each equation. Check your answers. $$ x^{2}+25=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2} + 25 = 0\) are \(x = 5i\) and \(x = -5i\).

Step 1: Identify the Equation Type

Recognize that the equation is a quadratic equation in the standard form of \(ax^{2} + bx + c = 0\), where \(a = 1\), \(b = 0\), and \(c = 25\).

Step 2: Solve for \(x\)

To find the values of \(x\) that satisfy the equation \(x^{2}+25=0\), subtract 25 from both sides to isolate the \(x^{2}\) term: \(x^{2} = -25\).

Step 3: Find the Square Roots

Solve for \(x\) by taking the square root of both sides of the equation. Since you cannot have a real number squared equal a negative in the real number system, this equation has no real solution. However, it has complex solutions: \(x = \pm i\sqrt{25}\).

Step 4: Simplify the Solutions

Simplify the square root to find the complex solutions: \(x = \pm 5i\).

Key Concepts

Complex Numbers

When solving quadratic equations, we sometimes find ourselves stumbling upon an interesting mathematical phenomenon: the square root of a negative number. Traditional arithmetic tells us that a negative number doesn’t have a real square root because no real number multiplied by itself gives a negative result. This is where complex numbers come into play.

Complex numbers are an expansion of the familiar real number system and include numbers that can be expressed in the form of a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying the equation i² = -1. They allow us to perform operations that involve the square roots of negative numbers, providing us with solutions to equations that otherwise would have none within the realm of real numbers.

The concept of complex numbers is not just an abstract mathematical idea; it's widely used in various fields such as electrical engineering, physics, and anywhere the notion of orthogonality or phase difference is required. Enabling us to visualize and calculate phenomena that are not easily represented on the one-dimensional number line of real numbers.

Quadratic Formula

Quadratic equations, which are of the form ax² + bx + c = 0, where a, b, and c are constants with a ≠ 0, can generally be solved by a variety of methods. However, the most comprehensive tool at our disposal is the quadratic formula. This formula states that the solutions for x can be found using
\[\[\begin{align*} x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \end{align*}\]\]
It covers all the bases, giving us the possible two solutions based on the discriminant b² - 4ac. The discriminant determines the nature of the roots:

• Positive discriminant: two distinct real roots,
• Zero discriminant: one real root, and
• Negative discriminant: two distinct complex roots.

When the discriminant is negative, we use the concept of complex numbers to express the solutions. The importance of the quadratic formula lies in its universality for solving any quadratic equation, making it a staple in algebra.

No Real Solution

In the context of quadratic equations, the phrase 'no real solution' can be misleading. It does not mean that the equation is unsolvable, rather it indicates that the roots of the equation are not real numbers, but complex ones. This is the case when our quadratic equation has a negative discriminant.

Returning to our initial exercise, the quadratic equation x² + 25 = 0 provides an excellent example. By attempting to isolate x², we find that x² = -25, which makes it impossible to have a real number solution because the square of a real number cannot be negative. Instead, we delve into the realm of complex numbers and express the solution as x = ± 5i.

This situation is a direct application of the quadratic formula, where the discriminant (b² - 4ac) would be negative. It is crucial for students to understand that such quadratic equations do have solutions; they just happen to be in complex form, expanding our understanding and ability to solve problems that arise in pure and applied mathematics.