Problem 55 - Elementary Algebra

Question

Solve using any method. $$ x 2-x-3=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1 + \sqrt{13}}{2} \) and \( x = \frac{1 - \sqrt{13}}{2} \).

Step 1: Recognize the Equation Type

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -1 \), and \( c = -3 \).

Step 2: Use the Quadratic Formula

Apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve the equation. Substitute \( a = 1 \), \( b = -1 \), and \( c = -3 \) into the formula.

Step 3: Calculate the Discriminant

First, calculate the discriminant \( b^2 - 4ac \). Substituting the values, we have \( (-1)^2 - 4(1)(-3) = 1 + 12 = 13 \). Since the discriminant is positive, there are two distinct real roots.

Step 4: Find the Roots

Using the discriminant in the quadratic formula gives \( x = \frac{-(-1) \pm \sqrt{13}}{2(1)} \). This simplifies to \( x = \frac{1 \pm \sqrt{13}}{2} \). Thus, the solutions are \( x = \frac{1 + \sqrt{13}}{2} \) and \( x = \frac{1 - \sqrt{13}}{2} \).

Key Concepts

Quadratic Formula

When solving quadratic equations, the Quadratic Formula is a powerful tool. It is applicable to any equation of the form \( ax^2 + bx + c = 0 \). This formula automatically gives us the solutions to the equation, often referred to as the "roots." The Quadratic Formula is expressed as:

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

Here, \( a \), \( b \), and \( c \) are coefficients in the quadratic equation.
Using this formula requires no additional factoring or guesswork, making it especially useful when dealing with complex or cumbersome expressions. Simply substitute the values of \( a \), \( b \), and \( c \) into the formula.
This formula considers every type of quadratic equation and can handle positive or negative coefficients, as well as special cases with complex roots.

Discriminant

The discriminant is a key part of the Quadratic Formula, located under the square root sign: \( b^2 - 4ac \). Its value tells us important information about the nature of the roots of a quadratic equation.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root (or a double root).
• If negative, the equation results in complex roots, not real numbers.

In our example, with \( a = 1 \), \( b = -1 \), and \( c = -3 \), the discriminant \( (-1)^2 - 4(1)(-3) \) results in 13, a positive number.
This positive discriminant indicates two distinct real roots, helping gauge the expected solutions before computing the actual roots.

Real Roots

Real roots are the actual solutions to a quadratic equation where the values are real numbers. When the discriminant is positive, as in our previous calculation, the quadratic equation will yield two real roots.
Using the quadratic formula, the plus-minus symbol (\( \pm \)) provides these two solutions:

• \( x = \frac{1 + \sqrt{13}}{2} \)
• \( x = \frac{1 - \sqrt{13}}{2} \)

These represent the points where the parabola, described by the quadratic equation, intersects the x-axis in a graph.
Real roots are crucial because they often represent physical or practical solutions to real-world problems. They help us understand phenomena in physics, engineering, and many other fields, where results are expected to be real and tangible.