Explanation And Solution For "X² + 5X + 10"

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Understanding the Problem

The given expression is a quadratic equation in the form of ax² + bx + c = 0, where a, b, and c are constants, and x is a variable. In this case, a = 1, b = 5, and c = 10. The goal is to find the solutions for x that satisfy the equation.

Using the Quadratic Formula

One way to solve a quadratic equation is by using the quadratic formula, which is x = (-b ± √(b² - 4ac)) / 2a. Substituting the values of a, b, and c, we get x = (-5 ± √(5² - 4(1)(10))) / 2(1), which simplifies to x = (-5 ± √(-15)) / 2. Since the square root of a negative number is not a real number, this equation has no real solutions.

Graphical Solution

Another way to understand the behavior of the quadratic equation is by graphing it. The graph of y = x² + 5x + 10 is a parabola that opens upward, with its vertex at (-5/2, 15/4). Since the vertex is above the x-axis, the equation has no real roots.

Completing the Square

A third method to solve the quadratic equation is by completing the square. To do this, we need to add and subtract (b/2)² inside the parentheses, which gives x² + 5x + (25/4) - (25/4) + 10. This can be rewritten as (x + 5/2)² - 15/4 + 10. Simplifying further, we get (x + 5/2)² + 5/4 = 0. Since the square of any real number is non-negative, this equation has no real solutions.

Imaginary Solutions

Although the equation x² + 5x + 10 has no real solutions, it does have two imaginary solutions. Imaginary numbers are in the form of a + bi, where a and b are real numbers, and i is the imaginary unit, which is defined as the square root of -1. The solutions for x can be expressed as (-5 ± √(-15)) / 2, which simplifies to (-5 ± i√15) / 2.

Factoring

A fifth method to solve the quadratic equation is by factoring. However, since the equation x² + 5x + 10 has no real roots, it cannot be factored into two linear factors with real coefficients.

Applications

Quadratic equations are used in many fields, such as physics, engineering, economics, and finance. For example, a quadratic equation can be used to find the maximum or minimum value of a function, the roots of a polynomial, or the trajectory of a projectile.

Conclusion

In summary, the quadratic equation x² + 5x + 10 has no real solutions, but it does have two imaginary solutions. This can be proven by using various methods, such as the quadratic formula, graphing, completing the square, and factoring. Understanding quadratic equations is essential for solving many practical problems in different fields.