What Are The Solutions To The Equation X2 6x 40

What Are the Solutions to the Equation x² + 6x + 40 = 0? A Deep Dive into Quadratic Equations

This article explores the solutions to the quadratic equation x² + 6x + 40 = 0, delving into the various methods for solving such equations and examining the nature of the solutions obtained. We'll cover the quadratic formula, completing the square, and graphical representations, providing a comprehensive understanding of this fundamental algebraic concept.

Understanding Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to a quadratic equation, also known as its roots or zeros, represent the values of x that satisfy the equation. These solutions can be real numbers or complex numbers.

Our specific equation, x² + 6x + 40 = 0, has a = 1, b = 6, and c = 40. Since we're dealing with a quadratic equation, we expect to find two solutions (although they may be identical).

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It provides a direct way to calculate the roots, regardless of their nature (real or complex). The formula is:

x = [-b ± √(b² - 4ac)] / 2a

Let's apply this formula to our equation:

• a = 1
• b = 6
• c = 40

Substituting these values into the quadratic formula, we get:

x = [-6 ± √(6² - 4 * 1 * 40)] / (2 * 1) x = [-6 ± √(36 - 160)] / 2 x = [-6 ± √(-124)] / 2

Since the discriminant (the value inside the square root, b² - 4ac) is negative (-124), the solutions are complex numbers involving the imaginary unit 'i', where i² = -1.

We can simplify the expression:

x = [-6 ± √(124)i] / 2 x = [-6 ± 2√(31)i] / 2 x = -3 ± √(31)i

Therefore, the solutions to the equation x² + 6x + 40 = 0 are:

• x₁ = -3 + √(31)i
• x₂ = -3 - √(31)i

These are two complex conjugate roots. Complex conjugate roots always appear in pairs when dealing with quadratic equations with real coefficients.

Method 2: Completing the Square

Completing the square is another method for solving quadratic equations. This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored.

Let's apply this method to our equation:

x² + 6x + 40 = 0

1. Move the constant term to the right side:

x² + 6x = -40

2. Take half of the coefficient of x (which is 6), square it (3² = 9), and add it to both sides:

x² + 6x + 9 = -40 + 9 x² + 6x + 9 = -31

3. Factor the left side as a perfect square trinomial:

(x + 3)² = -31

4. Take the square root of both sides:

x + 3 = ±√(-31) x + 3 = ±√(31)i

5. Solve for x:

x = -3 ± √(31)i

This yields the same solutions as the quadratic formula:

• x₁ = -3 + √(31)i
• x₂ = -3 - √(31)i

Method 3: Graphical Representation

A graphical representation can provide a visual understanding of the solutions. The solutions to the equation x² + 6x + 40 = 0 are the x-intercepts (points where the graph crosses the x-axis) of the parabola represented by the function y = x² + 6x + 40.

Since the solutions are complex numbers, the parabola will not intersect the x-axis. This is because the parabola opens upwards (since a = 1 > 0) and its vertex lies above the x-axis.

To find the vertex, we can use the formula for the x-coordinate of the vertex:

x = -b / 2a = -6 / (2 * 1) = -3

Substituting x = -3 into the equation, we find the y-coordinate:

y = (-3)² + 6(-3) + 40 = 9 - 18 + 40 = 31

Therefore, the vertex of the parabola is at (-3, 31). The parabola is entirely above the x-axis, confirming that there are no real solutions, only complex conjugate solutions.

Understanding the Discriminant

The discriminant (b² - 4ac) plays a crucial role in determining the nature of the solutions:

• If b² - 4ac > 0: The equation has two distinct real roots.
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex conjugate roots (as in our case).

Applications of Quadratic Equations

Quadratic equations have wide-ranging applications in various fields, including:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing stress and strain, optimizing system performance.
• Economics: Modeling supply and demand, analyzing market equilibrium.
• Computer Graphics: Creating curves and shapes, generating realistic images.

Conclusion

The equation x² + 6x + 40 = 0 has two complex conjugate roots: -3 + √(31)i and -3 - √(31)i. We explored three different methods—the quadratic formula, completing the square, and graphical representation—to solve this equation and understand the nature of its solutions. The discriminant was instrumental in predicting the type of roots before solving the equation. Understanding quadratic equations and their various solution methods is fundamental to various fields of study and practical applications. The methods presented here provide a comprehensive approach to tackling such equations, enhancing your problem-solving skills in algebra and beyond. Remember to always check your solutions by substituting them back into the original equation.