Problem 58 - Algebra A Combined Function

Question

Solve each equation by completing the square. $$ 4 x^{2}-2 x+5=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{4} \pm i \frac{\sqrt{19}}{4}\).

Step 1: Move the constant term

Begin by rearranging the equation to move the constant term to the right side. So, subtract \(5\) from both sides of the equation: \(4x^2 - 2x = -5\).

Step 2: Divide all terms by the leading coefficient

To simplify the quadratic term, divide every term in the equation by \(4\). The equation becomes \(x^2 - \frac{1}{2}x = -\frac{5}{4}\).

Step 3: Calculate and add the square

Take half of the coefficient of \(x\), which is \(-\frac{1}{2}\), divide it by \(2\), getting \(-\frac{1}{4}\). Square this value to obtain \(\left(-\frac{1}{4}\right)^2 = \frac{1}{16}\). Add and subtract this value inside the equation to maintain equality: \(x^2 - \frac{1}{2}x + \frac{1}{16} = -\frac{5}{4} + \frac{1}{16}\).

Step 4: Factor the perfect square trinomial

The left side of the equation is a perfect square trinomial and can be factored as \(\left(x - \frac{1}{4}\right)^2\). The equation is now \(\left(x - \frac{1}{4}\right)^2 = -\frac{5}{4} + \frac{1}{16}\).

Step 5: Simplify the right side

Simplify the expression on the right. Convert \(-\frac{5}{4}\) to \(-\frac{20}{16}\) so it can be easily added to \(\frac{1}{16}\). Thus, \(-\frac{20}{16} + \frac{1}{16} = -\frac{19}{16}\). The equation becomes \(\left(x - \frac{1}{4}\right)^2 = -\frac{19}{16}\).

Step 6: Solve by taking the square root

Take the square root of both sides. This yields: \(x - \frac{1}{4} = \pm \sqrt{-\frac{19}{16}}\). Since \(-\frac{19}{16}\) is negative, \(\sqrt{-\frac{19}{16}} = i \frac{\sqrt{19}}{4}\). Therefore, \(x - \frac{1}{4} = \pm i \frac{\sqrt{19}}{4}\).

Step 7: Solve for x

Add \(\frac{1}{4}\) to both sides to solve for \(x\). The solutions are \(x = \frac{1}{4} \pm i \frac{\sqrt{19}}{4}\). Thus, \(x = \frac{1}{4} + i \frac{\sqrt{19}}{4}\) or \(x = \frac{1}{4} - i \frac{\sqrt{19}}{4}\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that take the form \(ax^2 + bx + c = 0\). These equations are called "quadratic" because the highest degree of the variable \(x\) is squared (where "quad" means square). Each term in the equation can have different coefficients, and the main goal is to find the value(s) of \(x\) that make the equation true.

Quadratic equations are often solved using various methods such as factoring, completing the square, or applying the quadratic formula. Completing the square, as used in our original exercise, is a method where we adjust the equation into a perfect square trinomial on one side. This makes solving the equation more straightforward and helps reveal the nature of the solutions, especially when handling complex numbers.

The structure of these equations allows them to represent a simple parabola when graphed in a coordinate plane. The solutions to the equation correspond to the points where the parabola intersects the x-axis. Thus, understanding quadratic equations provides a foundation for analyzing real-world problems involving parabolic motion, and more.

Complex Numbers

Complex numbers are extensions of real numbers, typically represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, defined by the property \(i^2 = -1\).

In the context of solving quadratic equations, complex numbers become necessary when the discriminant, a part of the quadratic formula \(b^2 - 4ac\), yields a negative result. In such cases, solutions cannot be found among real numbers alone. For example, in our exercise, when the solutions end up involving a negative value under the square root, we need complex numbers to express the solution. Here, the result \(i \frac{\sqrt{19}}{4}\) involves the imaginary unit, representing an essential aspect of modern math used in fields as diverse as engineering and physics.

Complex solutions indicate a lack of real roots and suggest certain features in the graph of the quadratic function, like having a vertex point that does not touch the x-axis.

Factoring Trinomials

Factoring trinomials is a method used to simplify quadratic equations, making them easier to solve. A trinomial is a polynomial with three terms, typically structured in the form \(ax^2 + bx + c\).

In the process of completing the square, we effectively create a perfect square trinomial from the original equation. This manipulation relies on identifying parts of the equation that can be expressed as \((x+d)^2\), where \(d\) is derived from the original coefficients. By ensuring the quadratic and linear terms relate correctly, the trinomial factorization can be used to simplify solving for \(x\).

This important concept helps to understand the structure and behavior of quadratic equations, allowing for simplification and quicker resolution.

Solving Equations

Solving equations, particularly quadratics, involves finding all values of the variable that satisfy the equation. This process can take several forms, depending on the structure and context of the equation.

Completing the square is a targeted approach where we manipulate the quadratic to isolate the variable squared. By converting the problem into a more manageable equation \((x-h)^2 = k\), the solutions can be reached through square roots, thereby simplifying the process compared to other methods.

Other methods include using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which provides a direct solution to any quadratic. Factoring, as mentioned in previous sections, is also effective when applicable.

Each strategy caters to different complexities within equations and highlights the versatility inherent in algebra. Understanding these various methods enhances problem-solving skills and can be applied across different mathematical challenges.