Problem 81 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ 4 x^{2}-4 x=-1 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation: \(4x^2 - 4x = -1\) Answer: \(x = \frac{1}{2}\)

Step 1: Put the equation in standard form

Rewrite the equation in the standard form, which is \(ax^2 + bx + c = 0\). Currently, the equation is given as $$4x^2 - 4x = -1$$ Add 1 to both sides to set it equal to 0: $$4x^2 - 4x + 1 = 0$$

Step 2: Identify a, b, and c

From the equation in standard form, $$4x^2 - 4x + 1 = 0$$, identify the values of a, b, and c, which are the coefficients of the terms: $$a = 4$$ $$b = -4$$ $$c = 1$$

Step 3: Apply the quadratic formula

Now, use the quadratic formula to solve for x: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Plug in the values of a, b, and c: $$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)}$$

Step 4: Simplify the equation

Simplify the equation to find the solutions for x: $$x = \frac{4 \pm \sqrt{16 - 16}}{8}$$ $$x = \frac{4 \pm \sqrt{0}}{8}$$ $$x = \frac{4 \pm 0}{8}$$ $$x = \frac{4}{8}$$

Step 5: Find the solution

Simplify the fraction to find the solution for x: $$x = \frac{1}{2}$$ Thus, the solution for the given equation $$4x^2 - 4x = -1$$ is $$x = \frac{1}{2}$$.

Key Concepts

Standard Form

The standard form of a quadratic equation is a fundamental concept in mathematics. It is represented as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable to be solved. This form is essential because it allows us to easily apply methods like factoring, completing the square, or using the quadratic formula to find the solutions to the equation. When faced with a quadratic equation, the first goal is usually to rearrange it into standard form. Doing this sets up the equation for solving, making sure all terms are on one side and zero is on the other. For example, when given the equation \(4x^2 - 4x = -1\), we can add 1 to both sides to rewrite it as \(4x^2 - 4x + 1 = 0\). Now, it is in standard form, which is the starting point for many solution techniques.

Quadratic Formula

The quadratic formula is a reliable tool for solving quadratic equations that can be daunting by other methods. It is expressed as:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula helps find the values of \(x\) that make the equation true when the quadratic is in standard form. To use it, you need to substitute the coefficients \(a\), \(b\), and \(c\) from the equation into the formula. One of the powerful aspects of the quadratic formula is its ability to solve any quadratic equation, whether it can be factored easily or not. In our problem, after converting the equation to standard form, we plug in: \(a = 4\), \(b = -4\), \(c = 1\). After substitution, the equation becomes \(x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \times 4 \times 1}}{2 \times 4}\). Solving this simplifies to \(x = \frac{1}{2}\). The formula is particularly useful as it not only provides the solution but also helps determine the nature of the roots from \(b^2 - 4ac\), which is part of the discriminant.

Coefficients

Coefficients in a quadratic equation are the numerical or constant multipliers of the variables. At its core, understanding coefficients enables you to identify how each part of the quadratic equation contributes to the solution.

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term without \(x\)

In the equation \(4x^2 - 4x + 1 = 0\), these coefficients are \(a = 4\), \(b = -4\), and \(c = 1\). Recognizing these allows you to substitute them into the quadratic formula accurately, influencing the discriminant \(b^2 - 4ac\). The discriminant determines the nature of the roots, such as whether they are real or complex, or whether there is a single repeated root. In simple terms, coefficients help create the unique characteristics of each quadratic equation, directing the path to finding its solution.