Solving 4x² - 40x + 109 = 0 Find The Solutions

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In this article, we will delve into the process of finding the solutions to the quadratic equation 4x² - 40x + 109 = 0. Quadratic equations, which take the general form ax² + bx + c = 0, are fundamental in mathematics and have applications in various fields, including physics, engineering, and economics. Solving these equations involves determining the values of x that satisfy the equation, also known as the roots or zeros of the quadratic. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, we will employ the quadratic formula, a versatile method that works for any quadratic equation, regardless of whether it can be easily factored.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool derived from the method of completing the square. It provides a direct way to find the solutions of any quadratic equation in the form ax² + bx + c = 0. The formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Here, a, b, and c are the coefficients of the quadratic equation. The expression inside the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the solutions. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one real solution (a repeated root). If the discriminant is negative, the equation has two complex solutions. The complex solutions will always occur in conjugate pairs, meaning that if a + bi is a solution, then a - bi is also a solution, where i is the imaginary unit (√-1).

Applying the Quadratic Formula to Our Equation

Now, let's apply the quadratic formula to solve the equation 4x² - 40x + 109 = 0. First, we identify the coefficients:

  • a = 4
  • b = -40
  • c = 109

Next, we substitute these values into the quadratic formula:

x = (-(-40) ± √((-40)² - 4 * 4 * 109)) / (2 * 4)

Simplifying the expression, we get:

x = (40 ± √(1600 - 1744)) / 8

x = (40 ± √(-144)) / 8

Dealing with the Negative Discriminant

Notice that the discriminant is -144, which is negative. This indicates that the equation has complex solutions. We can rewrite the square root of -144 as:

√(-144) = √(144 * -1) = √144 * √-1 = 12i

where i is the imaginary unit, defined as √-1. Substituting this back into our equation, we get:

x = (40 ± 12i) / 8

Simplifying the Complex Solutions

We can simplify the solutions by dividing both the real and imaginary parts by 8:

x = (40/8) ± (12i/8)

x = 5 ± (3/2)i

Therefore, the solutions to the quadratic equation 4x² - 40x + 109 = 0 are:

x = 5 + (3/2)i and x = 5 - (3/2)i

These are complex conjugate solutions, as expected since the discriminant was negative. We can express the solutions in the form a + bi, where a is the real part and b is the imaginary part. In this case, the real part is 5 and the imaginary part is ±(3/2).

The Significance of the Discriminant

As we've seen, the discriminant (b² - 4ac) plays a vital role in determining the nature of the solutions of a quadratic equation. Let's explore this further:

  • If b² - 4ac > 0: The equation has two distinct real solutions. This means the parabola represented by the quadratic equation intersects the x-axis at two different points.
  • If b² - 4ac = 0: The equation has exactly one real solution (a repeated root). This means the parabola touches the x-axis at exactly one point, the vertex.
  • If b² - 4ac < 0: The equation has two complex solutions. This means the parabola does not intersect the x-axis at any real point. The solutions are complex conjugates, as we saw in our example.

Understanding the discriminant allows us to predict the type of solutions we will obtain before even applying the quadratic formula. This can be particularly useful in problem-solving and when analyzing the behavior of quadratic functions.

Methods for Solving Quadratic Equations

While the quadratic formula is a universal method, there are other techniques for solving quadratic equations, each with its advantages and disadvantages:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. Factoring is efficient when the roots are rational and the factors are easily identifiable. However, not all quadratic equations can be easily factored.
  2. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. Completing the square is useful for deriving the quadratic formula and can be applied even when factoring is not straightforward.
  3. Quadratic Formula: As discussed, this formula provides a direct solution for any quadratic equation. It is particularly useful when the equation is difficult to factor or when the solutions are irrational or complex.

The choice of method often depends on the specific equation and personal preference. However, the quadratic formula remains the most versatile and reliable approach for solving any quadratic equation.

Real-World Applications of Quadratic Equations

Quadratic equations are not just abstract mathematical concepts; they have numerous real-world applications in various fields:

  • Physics: Projectile motion, such as the trajectory of a ball thrown in the air, can be modeled using quadratic equations. The equation describes the height of the projectile as a function of time, considering factors like initial velocity and gravity.
  • Engineering: Quadratic equations are used in structural engineering to calculate the stresses and strains in beams and other structural components. They also appear in electrical engineering for analyzing circuits and determining impedance.
  • Economics: Quadratic functions can model cost, revenue, and profit curves in business and economics. Finding the maximum or minimum points of these curves, which often involves solving quadratic equations, is crucial for optimization problems.
  • Computer Graphics: Quadratic equations are used in computer graphics for creating curves and surfaces. Bezier curves, which are widely used in computer-aided design (CAD) and animation, are based on quadratic and cubic equations.

These are just a few examples of the many applications of quadratic equations. Their ability to model parabolic relationships makes them indispensable tools in science, technology, and everyday life.

Let's recap the step-by-step solution to the quadratic equation 4x² - 40x + 109 = 0:

  1. Identify the coefficients: a = 4, b = -40, c = 109.
  2. Apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).
  3. Substitute the values: x = (-(-40) ± √((-40)² - 4 * 4 * 109)) / (2 * 4).
  4. Simplify the expression: x = (40 ± √(1600 - 1744)) / 8.
  5. Calculate the discriminant: b² - 4ac = -144.
  6. Handle the negative discriminant: √(-144) = 12i.
  7. Substitute back into the equation: x = (40 ± 12i) / 8.
  8. Simplify the complex solutions: x = 5 ± (3/2)i.

Therefore, the solutions are x = 5 + (3/2)i and x = 5 - (3/2)i.

In conclusion, solving quadratic equations is a fundamental skill in mathematics with widespread applications. The quadratic formula provides a powerful and versatile method for finding the solutions, regardless of whether they are real or complex. Understanding the discriminant helps us predict the nature of the solutions, and various real-world examples demonstrate the importance of quadratic equations in diverse fields. By mastering the techniques discussed in this article, you can confidently tackle quadratic equations and appreciate their significance in mathematical problem-solving and beyond. The solutions to the quadratic equation 4x² - 40x + 109 = 0 are complex numbers, specifically x = 5 ± (3/2)i, highlighting the elegance and completeness of the complex number system in solving algebraic problems.