Problem 55 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^2 - 2x - 4 = 0\) are \(1 + \sqrt{5}\) and \(1 - \sqrt{5}\)

Step 1 - Identify the coefficients

From the quadratic equation \(x^2-2x-4=0\), we can identify the coefficients as: \(a=1\), \(b=-2\), \(c=-4\)

Step 2 - Plug in the coefficients to the formula

Substitute the values of a, b and c from our quadratic equation into the quadratic formula: \[x = \frac{-(-2) \pm \sqrt{(-2^2)-4*1*(-4)}}{2*1}\]

Step 3 - Simplify under the square root

Inside of the square root we have \((-2)^2 - 4*1*(-4)\) which is \(4+16\) and yields \(20\)

Step 4 - Simplify the equation completely

Simplify further to find the solutions to the equation: \[x=\frac{2\pm \sqrt{20}}{2}\]. Thus the solutions are \(x = 1 + \sqrt{5}\) and \(x = 1 - \sqrt{5}\)

Key Concepts

Quadratic Formula

Solving quadratic equations can be easily done using the quadratic formula. This formula is a powerful tool as it provides the solutions or "roots" for any quadratic equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• Here, \( a \), \( b \), and \( c \) are coefficients derived from the quadratic equation.
• The symbol \( \pm \) indicates that there can be two solutions.

Using the formula involves substituting the coefficients into the equation and calculating to find the roots. It streamlines the process by summarizing steps including completing the square and checking the discriminant.

Quadratic Coefficients

In any quadratic equation, the elements \( a \), \( b \), and \( c \) refer to the quadratic coefficients. These are the numerical factors preceding each term of the equation: \( ax^2 + bx + c = 0 \).

• Coefficient \( a \) is associated with the \( x^2 \) term. It is the key factor that appraises the equation's quadratic nature and influences the parabola's curvature direction (upwards if positive, downwards if negative).
• Coefficient \( b \) represents the linear term and impacts the parabola's symmetry.
• The constant term \( c \) acts merely as a vertical shift.

For example, in our specific equation \( x^2 - 2x - 4 = 0 \), the coefficients are \( a = 1 \), \( b = -2 \), and \( c = -4 \). Understanding these helps correctly substitute in the quadratic formula.

Discriminant

The discriminant is the element under the square root in the quadratic formula: \( b^2 - 4ac \). It plays a crucial role because it determines the nature of the roots.

• If the discriminant is positive, it implies that the quadratic equation has two distinct real roots.
• If it's zero, the equation has exactly one real (repeated) root.
• And if it's negative, the equation has two complex roots, indicating the solutions aren't real numbers.

For example, in the equation \( x^2 - 2x - 4 = 0 \), we calculated the discriminant as \( (-2)^2 - 4 \, * \, 1 \, * \, (-4) = 20 \). Since 20 is positive, the equation has two distinct real roots.

Roots of a Quadratic Equation

The roots of a quadratic equation are the solutions for which the equation \( ax^2 + bx + c = 0 \) equals zero. These roots can be found using several methods, including factoring, completing the square, or the most universal one — the quadratic formula.

• The term "roots" refers to the points on a graph where the parabola crosses the x-axis.
• Depending on the discriminant, there could be two distinct real roots, one real double root, or two complex roots.

In the provided example \( x^2 - 2x - 4 = 0 \), we found the roots as \( x = 1 + \sqrt{5} \) and \( x = 1 - \sqrt{5} \). These solutions mean that these are the x-values where the equation balances to zero, intersecting the x-axis.