Problem 55 - Algebra 1

Question

Solve the quadratic equation. $$4 x^{2}-2 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(x = [1 ± sqrt(5)] / 4\)

Step 1: Identify values

Identify the values of a, b, and c in the equation. In this case, \(a = 4\), \(b = -2\), and \(c = -1\).

Step 2: Apply the quadratic formula

Plug the values of a, b and c into the quadratic formula \(x = [-b ± sqrt(b ^2 - 4ac)] / 2a \). This leads to: \(x = [2 ± sqrt((-2) ^2 - 4*4*(-1))] / (2*4)\).

Step 3: Simplify the equation

Simplify the equation further: \(x = [2 ± sqrt(4 + 16)] / 8\). This simplifies to \(x = [2 ± sqrt(20)] / 8\).

Step 4: Calculate the final answer

Finally, simplify the above expression to get the roots of the quadratic. This results in two roots: \(x = [2 ± 2*sqrt(5)] / 8 = [1 ± sqrt(5)] / 4\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of a quadratic equation. A quadratic equation takes the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula is given as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to solve for \(x\), which represents the roots of the equation, by substituting the values of \(a\), \(b\), and \(c\) directly into the formula.

• The "±" sign indicates that there are usually two solutions, one for each sign.
• The part under the square root, \(b^2 - 4ac\), is called the discriminant and plays a crucial role in determining the nature of the roots.

Learning to use the quadratic formula effectively is essential for solving quadratic equations, as it provides a straightforward method to find the exact solutions.

Roots of Quadratic Equations

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Depending on the value of the discriminant \((b^2 - 4ac)\), a quadratic equation can have different types of roots:

• If the discriminant is positive, the equation has two distinct real roots.
• If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the equation has two complex roots.

In the example given in the exercise, the discriminant \(b^2 - 4ac = 4 + 16 = 20\) is positive, which means the quadratic equation has two distinct real roots. These roots are often expressed in simplified form, such as \(x = \frac{1 \pm \sqrt{5}}{4}\), showing the exact solution.

Solving Equations

Solving quadratic equations generally involves finding the values of \(x\) that make the equation true. There are several methods for solving quadratic equations, including:

• Factoring: This involves expressing the equation as a product of its factors, which is usually possible when the roots are rational.
• Completing the Square: This method involves manipulating the equation into a perfect square trinomial.
• Using the Quadratic Formula: Relevant for all types of quadratic equations, especially when factoring is not feasible.

While the quadratic formula is often the most direct method for obtaining exact solutions, understanding each method aids in developing a complete grasp of solving quadratic equations. Each technique can be useful in different scenarios, and practicing them enhances problem-solving skills in mathematics.

Mathematical Concepts

Quadratic equations encapsulate a range of mathematical concepts, including algebraic expressions, functions, and specific solution techniques. Grasping these concepts is vital for solving quadratic equations and applying them in diverse mathematical areas.
Understanding how to manipulate algebraic expressions ensures effective problem-solving strategies. Recognizing how these equations model certain physical phenomena or describe relationships helps in real-world applications.
Quadratic equations also serve as an introduction to more advanced mathematical areas like calculus and complex numbers. They help in enhancing logical reasoning and analytical skills, which are useful across various mathematical disciplines.
Mastering the concept of quadratic equations is a stepping stone towards understanding more intricate mathematical theories and applications, paving the way for progressive learning in mathematics.