Problem 15 - Elementary Algebra

Question

For the following problems, solve the equations using the quadratic formula. $$ x^{2}-2 x-3=0 $$

Step-by-Step Solution

Answer

Answer: The solutions for the given quadratic equation are \(x = 3\) and \(x = -1\).

Step 1: Identify a, b, and c

From the given quadratic equation, \(x^2 - 2x - 3 = 0\), we can identify the values of a, b, and c as: a = 1 b = -2 c = -3

Step 2: Apply the quadratic formula

Now, we apply the quadratic formula using the values a, b, and c: $$ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-3)}}{2(1)} $$

Step 3: Simplify the equation

Next, we simplify the equation to find the solutions for x: $$ x = \frac{2 \pm \sqrt{4 + 12}}{2} $$ $$ x = \frac{2 \pm \sqrt{16}}{2} $$

Step 4: Solve for x

Now we have: $$ x = \frac{2 \pm 4}{2} $$ This gives us two solutions for x: $$ x_1 = \frac{2+4}{2} = \frac{6}{2} = 3 $$ $$ x_2 = \frac{2-4}{2} = \frac{-2}{2} = -1 $$ So the solutions for the equation \(x^2 - 2x -3 = 0\) are \(x = 3\) and \(x = -1\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically written in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). To solve quadratic equations, we often use the quadratic formula. This formula provides an easy method to find the roots of the equation, or simply put, the values of \( x \) that satisfy the equation.

The quadratic formula is given by:

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

This formula helps in finding the solutions or roots of any quadratic equation, regardless of whether they are real or complex. In the quadratic formula:

• The term \( -b \pm \sqrt{b^2 - 4ac} \) determines the possible solutions.
• \( \pm \) indicates that there can be two solutions, one with addition and one with subtraction.

To use the formula effectively, identifying the correct values of \( a \), \( b \), and \( c \) from the quadratic equation is essential.

Quadratic Equation Coefficients

The coefficients \( a \), \( b \), and \( c \) in the quadratic equation are crucial because they determine the shape and position of the parabola represented by the equation \( ax^2 + bx + c = 0 \). Understanding these coefficients will allow you to use the quadratic formula successfully.

**Identifying Coefficients**

• In the equation \( x^2 - 2x - 3 = 0 \), the coefficients are identifiable as follows:

• \( a = 1 \), the coefficient of \( x^2 \).
• \( b = -2 \), the coefficient of \( x \).
• \( c = -3 \), the constant term.

Identifying these accurately is the first and crucial step in applying the quadratic formula.

Roots of Quadratic Equations

The 'roots' of a quadratic equation refer to the solutions of the equation; these are the values of \( x \) that make the equation \( ax^2 + bx + c = 0 \) true. Using the quadratic formula helps in finding these roots in a systematic way.

**Types of Roots**

• Real and Distinct: When \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• Real and Equal: When \( b^2 - 4ac = 0 \), the equation has exactly one real root (a repeated root).
• Complex: When \( b^2 - 4ac < 0 \), the equation has two complex roots, which are conjugates.

In our example, \( b^2 - 4ac = 4 + 12 = 16 > 0 \), which means the quadratic equation has two distinct real roots, \( x = 3 \) and \( x = -1 \). These roots are where the parabola intersects the x-axis.