Solving $4x^2 = 2x - 3$ Using The Quadratic Formula

In the realm of mathematics, quadratic equations hold a significant place, and mastering their solutions is a fundamental skill. Among the various methods available, the quadratic formula stands out as a powerful and versatile tool. This article delves into the application of the quadratic formula to solve the equation $4x^2 = 2x - 3$, providing a step-by-step guide and a comprehensive understanding of the underlying concepts.

Understanding Quadratic Equations

Before we dive into the solution, let's first establish a clear understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. The general form of a quadratic equation is expressed as:

$ax^2 + bx + c = 0$

where 'a', 'b', and 'c' are constants, and 'x' represents the variable we aim to solve for. The coefficients 'a', 'b', and 'c' play crucial roles in determining the nature and number of solutions. The coefficient 'a' cannot be zero, as this would reduce the equation to a linear form. The solutions to a quadratic equation are also known as its roots or zeros, representing the values of 'x' that satisfy the equation.

The Quadratic Formula: A Universal Solver

The quadratic formula is a mathematical expression that provides a direct method for finding the solutions (roots) of any quadratic equation. It is derived from the process of completing the square and offers a universal approach, regardless of the complexity of the coefficients. The formula is stated as follows:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where:

• 'x' represents the solutions (roots) of the quadratic equation.
• 'a', 'b', and 'c' are the coefficients of the quadratic equation in the standard form $ax^2 + bx + c = 0$.
• The symbol '±' indicates that there are two possible solutions, one with addition and one with subtraction.
• The expression inside the square root, $b^2 - 4ac$, is known as the discriminant, which provides valuable information about the nature of the roots.

Applying the Quadratic Formula to $4x^2 = 2x - 3$

Now, let's apply the quadratic formula to solve the given equation, $4x^2 = 2x - 3$. The first step is to rewrite the equation in the standard form $ax^2 + bx + c = 0$. To achieve this, we subtract $2x$ and add 3 to both sides of the equation, resulting in:

$4x^2 - 2x + 3 = 0$

Now, we can identify the coefficients:

• $a = 4$
• $b = -2$
• $c = 3$

With the coefficients identified, we can substitute these values into the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(4)(3)}}{2(4)}$

Simplifying the expression step-by-step:

1. Simplify the negative signs: $x = \frac{2 \pm \sqrt{(-2)^2 - 4(4)(3)}}{2(4)}$
2. Evaluate the exponent: $x = \frac{2 \pm \sqrt{4 - 4(4)(3)}}{2(4)}$
3. Perform the multiplication inside the square root: $x = \frac{2 \pm \sqrt{4 - 48}}{2(4)}$
4. Subtract inside the square root: $x = \frac{2 \pm \sqrt{-44}}{2(4)}$
5. Evaluate the denominator: $x = \frac{2 \pm \sqrt{-44}}{8}$

Dealing with the Negative Discriminant

Notice that we have a negative value under the square root, which means the solutions will be complex numbers. To proceed, we express the square root of -44 in terms of the imaginary unit 'i', where $i = \sqrt{-1}$.

$\sqrt{-44} = \sqrt{4 \cdot -11} = \sqrt{4} \cdot \sqrt{-11} = 2i\sqrt{11}$

Substituting this back into the equation:

$x = \frac{2 \pm 2i\sqrt{11}}{8}$

Simplifying the Solutions

We can simplify the solutions by dividing both the numerator and denominator by their greatest common factor, which is 2:

$x = \frac{1 \pm i\sqrt{11}}{4}$

Therefore, the solutions to the quadratic equation $4x^2 = 2x - 3$ are:

$x_1 = \frac{1 + i\sqrt{11}}{4}$

$x_2 = \frac{1 - i\sqrt{11}}{4}$

Expressing the Solution Set

The solution set is the collection of all solutions to the equation, enclosed in curly braces. In this case, the solution set is:

$\left\{ \frac{1 + i\sqrt{11}}{4}, \frac{1 - i\sqrt{11}}{4} \right\}$

These solutions are complex conjugates, which is a characteristic of quadratic equations with negative discriminants. Complex solutions always come in conjugate pairs, meaning they have the same real part but opposite imaginary parts.

The Discriminant and the Nature of Roots

As mentioned earlier, the discriminant ($b^2 - 4ac$) provides valuable insights into the nature of the roots of a quadratic equation. There are three possible scenarios:

1. Positive Discriminant ($b^2 - 4ac > 0$): The equation has two distinct real roots.
2. Zero Discriminant ($b^2 - 4ac = 0$): The equation has one real root (a repeated root).
3. Negative Discriminant ($b^2 - 4ac < 0$): The equation has two complex roots (complex conjugates).

In our example, the discriminant is $(-2)^2 - 4(4)(3) = 4 - 48 = -44$, which is negative. This confirms that the equation has two complex roots, as we found in our solution.

Conclusion

The quadratic formula is a powerful tool for solving quadratic equations, regardless of the complexity of the coefficients. By following the step-by-step process outlined in this article, you can confidently solve any quadratic equation and determine the nature of its roots. Remember to rewrite the equation in standard form, identify the coefficients, substitute them into the formula, and simplify the resulting expression. Understanding the discriminant allows you to predict the type of solutions you will obtain, further enhancing your problem-solving skills in mathematics. With practice and a solid grasp of the quadratic formula, you'll be well-equipped to tackle a wide range of mathematical challenges.

Key Takeaways:

• The quadratic formula provides a direct method for solving quadratic equations.
• The discriminant ($b^2 - 4ac$) reveals the nature of the roots.
• A negative discriminant indicates complex roots.
• Complex roots always occur in conjugate pairs.
• The solution set encompasses all solutions to the equation.

By mastering the quadratic formula and its applications, you'll gain a deeper understanding of quadratic equations and their significance in mathematics and various fields of science and engineering. This knowledge will serve as a strong foundation for further exploration of mathematical concepts and problem-solving techniques.