Finding The Sum Of Absolute Values Of Roots In Quadratic Equations X^2 - 2x + 4|-5| = 5 - 3x
To find the sum of the absolute values of the roots of the equation X^2 - 2x + 4|-5| = 5 - 3x, we need to solve the equation and then calculate the sum of the absolute values of the solutions. Let's break down the process step by step.
1. Simplify the Equation
First, let's simplify the equation. We have the equation:
X^2 - 2x + 4|-5| = 5 - 3x
The absolute value of -5, denoted as |-5|, is 5. So, the equation becomes:
X^2 - 2x + 4(5) = 5 - 3x
Further simplifying:
X^2 - 2x + 20 = 5 - 3x
2. Rearrange the Equation
Next, we need to rearrange the equation into the standard quadratic form, which is Ax^2 + Bx + C = 0. To do this, we move all terms to one side of the equation:
X^2 - 2x + 20 - 5 + 3x = 0
Combining like terms, we get:
X^2 + x + 15 = 0
3. Solve the Quadratic Equation
Now, we have a quadratic equation in the standard form. We can solve it using the quadratic formula, which is:
X = [-B ± √(B^2 - 4AC)] / (2A)
In our equation, A = 1, B = 1, and C = 15. Plugging these values into the quadratic formula, we get:
X = [-1 ± √(1^2 - 4(1)(15))] / (2(1))
X = [-1 ± √(1 - 60)] / 2
X = [-1 ± √(-59)] / 2
Since the discriminant (the value inside the square root) is negative, we have complex roots. The roots are:
X = [-1 ± i√59] / 2
So, the two roots are:
X₁ = (-1 + i√59) / 2
X₂ = (-1 - i√59) / 2
4. Find the Absolute Values of the Roots
To find the absolute value of a complex number a + bi, we use the formula:
|a + bi| = √(a^2 + b^2)
For X₁ = (-1 + i√59) / 2, we have a = -1/2 and b = √59 / 2. Thus,
|X₁| = √((-1/2)^2 + (√59 / 2)^2)
|X₁| = √(1/4 + 59/4)
|X₁| = √(60/4)
|X₁| = √15
For X₂ = (-1 - i√59) / 2, we have a = -1/2 and b = -√59 / 2. Thus,
|X₂| = √((-1/2)^2 + (-√59 / 2)^2)
|X₂| = √(1/4 + 59/4)
|X₂| = √(60/4)
|X₂| = √15
5. Calculate the Sum of the Absolute Values
Finally, we add the absolute values of the roots:
Sum = |X₁| + |X₂|
Sum = √15 + √15
Sum = 2√15
Therefore, the sum of the absolute values of the roots of the given equation is 2√15.
In summary, to find the sum of the absolute values of the roots, we first simplified the equation by evaluating the absolute value term. Then, we rearranged the equation into the standard quadratic form and used the quadratic formula to find the roots. Because the discriminant was negative, the roots were complex. We calculated the absolute values of these complex roots using the formula |a + bi| = √(a^2 + b^2) and finally summed these absolute values to arrive at the solution. This process demonstrates a comprehensive approach to solving quadratic equations with complex roots and finding the sum of their absolute values.
In the previous section, we derived that the roots of the given equation are complex, specifically X₁ = (-1 + i√59) / 2 and X₂ = (-1 - i√59) / 2. Understanding complex roots is essential in various fields of mathematics and physics. Complex roots occur when the discriminant (B^2 - 4AC) of a quadratic equation is negative. This indicates that the parabola represented by the quadratic equation does not intersect the x-axis, meaning there are no real solutions.
The significance of complex roots extends beyond pure mathematics. In electrical engineering, for instance, complex numbers are used to represent alternating current (AC) circuits, where the imaginary part corresponds to the phase difference. In quantum mechanics, complex numbers are fundamental in describing wave functions and the behavior of particles. Therefore, mastering the manipulation and interpretation of complex roots is a crucial skill for anyone working in these fields.
In our specific example, the presence of complex roots tells us that there are no real solutions to the equation X^2 + x + 15 = 0. This means that there is no real number that, when substituted into the equation, will make it equal to zero. Instead, the solutions lie in the complex plane, which extends the real number line to include imaginary numbers.
When we calculated the absolute values of the complex roots, we were essentially finding the magnitude or modulus of these complex numbers. The absolute value of a complex number represents its distance from the origin in the complex plane. In our case, both roots have the same absolute value, √15, which means they are equidistant from the origin.
Furthermore, the fact that the roots are complex conjugates (i.e., they have the form a + bi and a - bi) is a common property of quadratic equations with real coefficients. Complex conjugate roots always have the same absolute value, which is why |X₁| and |X₂| were equal in our calculation. This symmetry is a direct consequence of the quadratic formula and the nature of complex numbers.
The final step of adding the absolute values of the roots gives us a measure of the total magnitude of the roots. This value, 2√15, can be interpreted as the sum of the distances of the roots from the origin in the complex plane. It provides a single number that summarizes the overall scale of the solutions to the equation.
In conclusion, the presence of complex roots in a quadratic equation is not just a mathematical curiosity but a meaningful indication of the equation's behavior and the nature of its solutions. Understanding complex roots, their absolute values, and their properties is crucial for a deeper understanding of algebra and its applications in various scientific and engineering disciplines. The detailed calculation and interpretation presented here underscore the importance of mastering these concepts.
The original equation, X^2 - 2x + 4|-5| = 5 - 3x, involves an absolute value term. Solving equations with absolute values often requires a different approach compared to standard algebraic equations. The primary reason for this is the piecewise nature of the absolute value function, which is defined as |x| = x if x ≥ 0 and |x| = -x if x < 0.
In our specific case, the absolute value term |-5| simplifies directly to 5, making the equation a standard quadratic equation once rearranged. However, when dealing with absolute value expressions involving variables, such as |x - a|, the problem becomes more complex. To tackle such equations, we typically need to consider different cases based on the sign of the expression inside the absolute value.
For example, consider an equation like |x - 3| = 2. To solve this, we need to consider two cases:
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Case 1: x - 3 ≥ 0 (i.e., x ≥ 3) In this case, |x - 3| = x - 3, so the equation becomes x - 3 = 2. Solving for x, we get x = 5. Since 5 ≥ 3, this solution is valid.
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Case 2: x - 3 < 0 (i.e., x < 3) In this case, |x - 3| = -(x - 3), so the equation becomes -(x - 3) = 2. This simplifies to -x + 3 = 2, and solving for x, we get x = 1. Since 1 < 3, this solution is also valid.
Therefore, the solutions to the equation |x - 3| = 2 are x = 1 and x = 5.
This casework approach is fundamental for solving more complex equations involving absolute values. For instance, if we have an equation like |2x + 1| = |x - 2|, we would need to consider four cases:
- 2x + 1 ≥ 0 and x - 2 ≥ 0
- 2x + 1 ≥ 0 and x - 2 < 0
- 2x + 1 < 0 and x - 2 ≥ 0
- 2x + 1 < 0 and x - 2 < 0
Each case leads to a different linear equation, and we solve each one separately, checking whether the solutions satisfy the initial conditions for that case.
In the context of quadratic equations with absolute values, the same principles apply. If we encounter an equation like |x^2 - 4| = 3x, we would need to consider the cases where x^2 - 4 ≥ 0 and x^2 - 4 < 0. This would involve solving different quadratic equations for each case and verifying the solutions.
Moreover, when graphing functions involving absolute values, the casework approach helps in understanding the shape of the graph. The absolute value function creates a “V” shape at the point where the expression inside the absolute value changes sign. This can be useful in visualizing the solutions to equations involving absolute values.
In summary, solving equations with absolute values requires a careful consideration of different cases based on the sign of the expression inside the absolute value. This technique, combined with the principles of solving linear and quadratic equations, provides a robust approach to tackling a wide range of problems involving absolute values. The step-by-step analysis and casework ensure that all possible solutions are identified and validated, leading to a comprehensive understanding of the equation's behavior.
Quadratic equations, like the one we solved (X^2 + x + 15 = 0), may seem abstract, but they have numerous real-world applications across various disciplines. Understanding these applications helps to appreciate the practical significance of quadratic equations beyond theoretical mathematics.
One of the most common applications is in physics, particularly in the study of projectile motion. The trajectory of a projectile, such as a ball thrown into the air, can be modeled using a quadratic equation. The equation describes the height of the projectile as a function of time, taking into account the initial velocity, launch angle, and the acceleration due to gravity. By solving a quadratic equation, we can determine the maximum height reached by the projectile, the time it takes to reach that height, and the total time the projectile is in the air.
For example, if we launch a ball with an initial upward velocity of v₀ and neglect air resistance, the height h(t) of the ball at time t can be described by the equation:
h(t) = -½gt² + v₀t + h₀
where g is the acceleration due to gravity (approximately 9.8 m/s²) and h₀ is the initial height. This equation is a quadratic in t, and solving it for h(t) = 0 gives the time when the ball hits the ground.
Another application of quadratic equations is in engineering, particularly in the design of parabolic structures such as bridges and antennas. The shape of a parabola, which is the graph of a quadratic function, is ideal for these structures because it can efficiently distribute loads and focus signals. For instance, the cables of a suspension bridge often form a parabolic shape, and the equation describing this shape is quadratic.
In economics and finance, quadratic equations are used to model cost, revenue, and profit functions. For example, the cost of producing a certain number of items may be modeled as a quadratic function, with the minimum cost occurring at the vertex of the parabola. Similarly, the profit function, which is the difference between revenue and cost, can also be quadratic. By finding the roots and vertex of the profit function, businesses can determine the break-even points and the production level that maximizes profit.
Quadratic equations also appear in computer graphics and game development. The paths of objects moving in a game environment, such as projectiles or characters jumping, are often described using quadratic equations. These equations allow developers to create realistic and visually appealing animations.
In geometry, quadratic equations are used to solve problems involving areas and distances. For example, if we know the area of a rectangle and the relationship between its length and width, we can set up a quadratic equation to find the dimensions of the rectangle. Similarly, the Pythagorean theorem, which relates the sides of a right triangle, leads to a quadratic equation when solving for an unknown side length.
Furthermore, in optimization problems, quadratic equations are used to find the maximum or minimum values of certain quantities. These problems often involve finding the vertex of a parabola, which corresponds to the maximum or minimum point of a quadratic function.
In conclusion, the applications of quadratic equations are vast and varied, spanning across physics, engineering, economics, computer science, and geometry. Their ability to model projectile motion, parabolic structures, economic functions, and geometric relationships makes them an indispensable tool in many fields. Understanding quadratic equations and their properties is therefore essential for anyone pursuing studies or careers in these areas. The ability to solve and interpret quadratic equations provides a powerful means of analyzing and solving real-world problems.