Problem 64 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}-2 x=2 $$

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are \(x = 1+sqrt(3)\) and \(x = 1-sqrt(3)\)

Step 1: Understand the Given Equation

The given equation is \(x^{2}-2x-2 = 0\). This is a quadratic equation which has the form \(ax^2 + bx + c = 0\). Here, a = 1, b = -2, and c = -2.

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = [-b ± sqrt(b^2-4ac)] / 2a\). Substitute the coefficients a, b, and c into the quadratic formula. This gives \(x = [2 ± sqrt((-2)^2 - 4*1*(-2))] / 2*1 = [2 ± sqrt(4 + 8)] / 2 = [2 ± sqrt(12)] / 2 = [2 ± 2sqrt(3)] / 2\).

Step 3: Simplify the solutions

When we simplify this, we get two solutions: \(x = 1+sqrt(3)\) and \(x = 1-sqrt(3)\)

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool in solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This formula allows us to find the solutions, also known as the roots, of the quadratic equation. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s how it works:

• The term \( -b \pm \sqrt{b^2 - 4ac} \) represents the calculation that leads to the potential solutions of the equation.
• The portion \( b^2 - 4ac \) is called the discriminant and determines the nature of the roots.
• Using \( 2a \) in the denominator dictates how these solutions are balanced through dividing, adding or subtracting the other values.

This formula is highly reliable for any quadratic and does not depend on other methods like factoring, which might not always be possible.

Solutions of Quadratic Equations

Quadratic equations typically have two solutions, which can often be real or complex numbers, depending on the discriminant \( b^2 - 4ac \). Here’s how to interpret these solutions:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \), both solutions are real and equal, leading to one repeated solution.
• If \( b^2 - 4ac < 0 \), the solutions are complex or imaginary numbers, involving the square root of a negative number.

In the equation \( x^2 - 2x - 2 = 0 \), using the quadratic formula gives us two real solutions because the discriminant is positive. After substituting into the formula, the solutions calculated are:

• \( x = 1 + \sqrt{3} \)
• \( x = 1 - \sqrt{3} \)

These solutions effectively satisfy the original quadratic equation when plugged back in.

Simplification of Solutions

Simplifying the results from the quadratic formula is the final step to reach the solutions. It involves careful arithmetic manipulation:
With the solutions \( x = \frac{2 \pm 2\sqrt{3}}{2} \), we perform simplification:

• First, note the common factor in the expressions, in this case, 2 in the numerator and denominator.
• By factoring, divide both the numerator and denominator by 2, resulting in \( x = 1 \pm \sqrt{3} \).

Here are some rules to bear in mind during simplification:

• Always look for common factors that can cancel out terms in the expression.
• Recognize that algebraic manipulation should not alter the magnitudes and directions of calculations but simplify the representation of the roots.

Simplification ensures the solutions are presented in their simplest and clearest form, making them easier to interpret and verify.