Problem 74 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solutions for the equation are \(x = 1 + \sqrt{5}\) and \(x = 1 - \sqrt{5}\).

Step 1: Identify a, b, and c

Here the equation is \(x^{2}-2 x -4 = 0\). So, from the equation we identify \(a = 1\), the coefficient of \(x^2\), \(b = -2\), the coefficient of x, and \(c = -4\), the constant term.

Step 2: Substitute into the Quadratic Formula

Substitute a, b, and c into the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Thus we have \(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4*1*(-4)}}{2*1}\). Simplifying this we get \(x = \frac{2 \pm \sqrt{20}}{2}\)

Step 3: Simplifying Further

Simplify the equation further. The square root of 20 simplifies to \(2\sqrt{5}\). Thus we have \(x = 1 \pm \sqrt{5}\)

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool in algebra when it comes to solving quadratic equations. It is a formula that provides the solutions to a quadratic equation of the form \(ax^2 + bx + c = 0\). This formula is

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

The symbols \(\pm\) indicate two possible solutions, which is typical for these equations. The part under the square root sign, \(b^2 - 4ac\), is called the discriminant. It tells us about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root.
• A negative discriminant indicates complex roots.

For our original exercise equation, \(x^2 - 2x - 4 = 0\), by identifying the values \(a = 1\), \(b = -2\), and \(c = -4\), we can successfully utilize the quadratic formula to find the roots.

Solving Quadratic Equations

Solving quadratic equations effectively involves following a series of logical steps. The typical methods include factorization, completing the square, and using the quadratic formula. For quadratic equations that are not easily factorable, the quadratic formula is often the most straightforward and reliable approach.

Starting with the equation \(x^2 - 2x - 4 = 0\), identifying \(a\), \(b\), and \(c\) is the first step. These coefficients are crucial for substituting into the quadratic formula. Upon substituting, we simplify and solve for \(x\).

• Use: \(x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \times 1 \times (-4)}}{2 \times 1}\)
• Simplify to solve for \(x\).

This systematic substitution and simplification reflect the core process needed to handle quadratic equations with confidence.

Algebraic Simplification

Algebraic simplification is the step where we "clean up" our math expressions. It includes simplifying the expression inside the square root, which often involves recognizing perfect squares or using basic arithmetic to reduce numbers.

In this example, simplifying \(\sqrt{20}\) is crucial. We break it down into its simpler components: write \(\sqrt{20} = \sqrt{4 \times 5}\). Recognizing that \(\sqrt{4} = 2\), we can then express it as \(2\sqrt{5}\). It is crucial at this stage to observe how simplification can change the complexity of a problem, making it easier and less error-prone.

• Result: \(x = 1 \pm \sqrt{5}\)

This gives you the most simplified form of the solution, making sure the answer is as clear and concise as possible.