Problem 25 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ 4 x^{2}-2 x-1=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given quadratic equation are $$x_1 = \frac{1 + \sqrt{5}}{4}$$ and $$x_2 = \frac{1 - \sqrt{5}}{4}$$.

Step 1: Identify a, b, and c

In the given equation, $$4x^2 - 2x - 1 = 0$$, we can see that: - a = 4 - b = -2 - c = -1

Step 2: Plug the values of a, b, and c into the quadratic formula

Now, we will use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Substitute the values of a, b, and c: $$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(4)(-1)}}{2(4)}$$

Step 3: Simplify the expression under the square root

Calculate the value inside the square root: $$(-2)^2 - 4(4)(-1) = 4 + 16 = 20$$ Now the equation becomes: $$x = \frac{2 \pm \sqrt{20}}{8}$$

Step 4: Simplify the expression

The square root of 20 can also be expressed as $$\sqrt{4 \cdot 5} = 2 \sqrt{5}$$. So the equation becomes: $$x = \frac{2 \pm 2\sqrt{5}}{8}$$

Step 5: Factor out the common factor

We can see that 2 is a common factor in both the numerator and the denominator, so let's divide both by 2: $$x = \frac{1 \pm \sqrt{5}}{4}$$ Now we have the two solutions for x: $$x_1 = \frac{1 + \sqrt{5}}{4}$$ $$x_2 = \frac{1 - \sqrt{5}}{4}$$

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable to solve for. These equations take on a U-shaped graph known as a parabola.

To find the roots, or solutions of a quadratic equation, the quadratic formula is a powerful tool. The formula is written as:

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

The symbol \(\pm\) represents the two possible solutions that arise because the parabola crosses the x-axis at two points in most cases.

Consider our example, \(4x^2 - 2x - 1 = 0\). Here, \(a = 4\), \(b = -2\), and \(c = -1\).

We substitute these values into the quadratic formula and simplify to find the roots.

Algebra

Algebra is a broad field of mathematics dealing with symbols and rules used to manipulate those symbols. It forms the foundation for solving equations like the quadratic ones.

When working with quadratic equations, identifying coefficients is part of basic algebra. Here’s what you typically do:

• Identify coefficients \(a\), \(b\), and \(c\) in the standard form of the equation \(ax^2 + bx + c = 0\).
• Substitute these into mathematical formulas.

In our specific equation, \(4x^2 - 2x - 1 = 0\), we break down algebraic steps:

• Introducing them into the quadratic formula step-by-step.
• Simplifying complex expressions such as \(\sqrt{b^2 - 4ac}\).

These operations allow us to engage in deeper understanding beyond just arithmetic, fostering a grasp of structure in mathematical equations.

Mathematical Problem Solving

Mathematical problem solving involves strategies and methods to find solutions. Solving quadratic equations requires logical reasoning and step-by-step processes.

The structured approach helps make sure you don't overlook critical details:

• First, extract \(a\), \(b\), and \(c\) from the equation.
• Next, input these into the quadratic formula.
• Resolve the discriminant \((b^2 - 4ac)\), which determines the nature of the roots (real or complex).

This method is a mental workout, strengthening analytical skills.

Once calculated, you simplify the equation as demonstrated:

• The intermediate step was finding \(\sqrt{20}\).
• Simplified to \(2\sqrt{5}\).
• Then, factor the expression to find the final roots: \(x_1\) and \(x_2\).

This structured problem-solving helps make sense of mathematical complexities, teaching a valuable process on how to break down problems effectively.