What Are The Solutions To The Equation X2 6x 40

Solving the Quadratic Equation: x² + 6x - 40 = 0

The quadratic equation x² + 6x - 40 = 0 is a fundamental problem in algebra. Understanding its solution reveals core concepts applicable to a wide range of mathematical problems. This comprehensive guide explores various methods to solve this equation, delving into the underlying principles and offering a detailed explanation of each approach. We'll cover factoring, the quadratic formula, completing the square, and graphical methods, providing a holistic understanding of this seemingly simple yet powerful equation.

Understanding Quadratic Equations

Before diving into the solutions, let's establish a basic understanding of quadratic equations. A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. Our specific equation, x² + 6x - 40 = 0, fits this form with a = 1, b = 6, and c = -40. The solutions to a quadratic equation are the values of 'x' that satisfy the equation, also known as the roots or zeros of the equation. These roots can be real numbers or complex numbers.

Method 1: Factoring

Factoring is a powerful method for solving quadratic equations, particularly when the equation is easily factorable. This involves expressing the quadratic expression as a product of two linear expressions.

The goal is to find two numbers that add up to 'b' (6 in our case) and multiply to 'c' (-40). Let's consider the factors of -40:

• 1 and -40
• 2 and -20
• 4 and -10
• 5 and -8
• -1 and 40
• -2 and 20
• -4 and 10
• -5 and 8

Notice that 10 and -4 satisfy our conditions: 10 + (-4) = 6 and 10 * (-4) = -40.

Therefore, we can factor the quadratic equation as:

(x + 10)(x - 4) = 0

This equation is satisfied if either (x + 10) = 0 or (x - 4) = 0. Solving these linear equations gives us the solutions:

• x + 10 = 0 => x = -10
• x - 4 = 0 => x = 4

Therefore, the solutions to the equation x² + 6x - 40 = 0 using factoring are x = -10 and x = 4.

Method 2: The Quadratic Formula

The quadratic formula is a universal method for solving any quadratic equation, regardless of its factorability. The formula is derived from completing the square (explained in the next section) and provides the solutions directly:

x = [-b ± √(b² - 4ac)] / 2a

Substituting the values from our equation (a = 1, b = 6, c = -40):

x = [-6 ± √(6² - 4 * 1 * -40)] / (2 * 1) x = [-6 ± √(36 + 160)] / 2 x = [-6 ± √196] / 2 x = [-6 ± 14] / 2

This gives us two solutions:

• x = (-6 + 14) / 2 = 8 / 2 = 4
• x = (-6 - 14) / 2 = -20 / 2 = -10

Therefore, using the quadratic formula, the solutions are x = 4 and x = -10. Note that this method confirms the solutions obtained through factoring.

Method 3: Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in a perfect square trinomial form. This method is particularly useful when factoring is difficult or impossible.

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 = 49

3. Rewrite the left side as a perfect square trinomial: (x + 3)² = 49

4. Take the square root of both sides: x + 3 = ±√49 x + 3 = ±7

5. Solve for 'x': x = -3 + 7 = 4 x = -3 - 7 = -10

Therefore, completing the square yields the solutions x = 4 and x = -10.

Method 4: Graphical Method

A graphical method involves plotting the quadratic function y = x² + 6x - 40 and finding the x-intercepts, which represent the solutions to the equation. The x-intercepts are the points where the graph intersects the x-axis (where y = 0).

While a precise graphical solution requires graphing tools or software, we can conceptually understand this method. The parabola represented by the quadratic equation will intersect the x-axis at two points, corresponding to the roots x = 4 and x = -10. These points represent where the value of the function (y) is zero, satisfying the original equation.

Understanding the Discriminant (b² - 4ac)

The expression b² - 4ac within the quadratic formula is called the discriminant. It provides valuable information about the nature of the roots:

• If b² - 4ac > 0: The equation has two distinct real roots. This is the case in our example (196 > 0).
• If b² - 4ac = 0: The equation has one real root (a repeated root).
• If b² - 4ac < 0: The equation has two complex conjugate roots (involving imaginary numbers).

Applications of Quadratic Equations

Quadratic equations are not just theoretical exercises; they have numerous applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.
• Engineering: Designing bridges, buildings, and other structures.
• Economics: Modeling supply and demand, analyzing market trends.
• Computer Graphics: Creating curves and shapes.

Conclusion

Solving the quadratic equation x² + 6x - 40 = 0 demonstrates several fundamental algebraic techniques. Factoring, the quadratic formula, completing the square, and graphical methods all lead to the same solutions: x = 4 and x = -10. Understanding these methods provides a strong foundation for tackling more complex mathematical problems and appreciating the practical applications of quadratic equations in various disciplines. The discriminant helps predict the nature of the solutions before even solving the equation, adding another layer of understanding. Mastering these concepts is crucial for success in higher-level mathematics and related fields.