Problem 52 - Precalculus Student Solutions Manual 5th

Question

Solve each equation using the quadratic formula. $$x^{2}-3 x-2=0$$

Step-by-Step Solution

Answer

The solutions are \[ x = \frac{3 + \sqrt{17}}{2} \] and \[ x = \frac{3 - \sqrt{17}}{2} \].

Step 1: Identify coefficients

Identify the coefficients from the quadratic equation in the form \[ ax^2 + bx + c = 0 \] For the given equation \[ x^2 - 3x - 2 = 0 \], the coefficients are: a = 1 b = -3 c = -2.

Step 2: Write the quadratic formula

The quadratic formula for solving equations of the form \[ ax^2 + bx + c = 0 \] is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 3: Substitute the coefficients into the formula

Substitute the coefficients \( a = 1 \), \( b = -3 \), and \( c = -2 \) into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-2)}}{2(1)} \] which simplifies to: \[ x = \frac{3 \pm \sqrt{9 + 8}}{2} \].

Step 4: Simplify under the square root

Simplify the expression under the square root: \[ 9 + 8 = 17 \], giving \[ x = \frac{3 \pm \sqrt{17}}{2} \].

Step 5: Find the roots

Express the two potential solutions separately: \[ x = \frac{3 + \sqrt{17}}{2} \] and \[ x = \frac{3 - \sqrt{17}}{2} \].

Key Concepts

Quadratic Equations

A quadratic equation is any polynomial equation of the form \[ ax^2 + bx + c = 0 \]. Here, \( a, b, \) and \( c \) are constants, with \( a eq 0 \). Quadratic equations are called so because they involve terms up to the second degree (\( x^2 \)). These equations often appear in various contexts, such as physics, engineering, and economics.

The standard form of a quadratic equation is important because it allows us to use the quadratic formula to find solutions or ‘roots’. For example, in the equation \[ x^2 - 3x - 2 = 0 \], \( x^2 \) is the quadratic term, \( -3x \) is the linear term, and \( -2 \) is the constant term. Understanding how to recognize and work with these terms is the first step to solving any quadratic equation.

Coefficients

Coefficients are the numerical values assigned to the variables in an equation. In a quadratic equation \[ ax^2 + bx + c = 0 \], the coefficient \( a \) is associated with the \( x^2 \) term, \( b \) with the \( x \) term, and \( c \) with the constant term.

For the equation \[ x^2 - 3x - 2 = 0 \], we identify our coefficients as:

• \( a = 1 \)
• \( b = -3 \)
• \( c = -2 \)

These coefficients are essential when using the quadratic formula, as you must substitute them correctly to solve for the variable \( x \). Remember, these coefficients can be positive or negative, affecting the final solution.

Roots of a Polynomial

The roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. In other words, they are the solutions to the equation.

In our example \[ x^2 - 3x - 2 = 0 \], the roots are the \( x \) values that satisfy this equation.
The quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] helps us find these roots. By substituting our identified coefficients, \( a = 1 \), \( b = -3 \), and \( c = -2 \), we get:
\ x = \frac{-(-3)\pm\sqrt{(-3)^2 - 4(1)(-2)}}{2(1)} \
Simplifying further:

• Under the square root: \( 9 + 8 = 17 \)
• So, \[ x = \frac{3 \pm \sqrt{17}}{2} \]

Which results in two roots:
\[ x = \frac{3 + \sqrt{17}}{2} \]
\[ x = \frac{3 - \sqrt{17}}{2} \]

Solving Equations

Solving quadratic equations often involves finding the points where the graph intersects the x-axis, which are the roots.

Here's a summarized process using the quadratic formula. First, identify the coefficients from the equation in the format \[ ax^2 + bx + c = 0 \]. Next, substitute these coefficients into the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

• Step 1: Write the quadratic formula.
• Step 2: Substitute your coefficients \( a, b, \) and \( c \).
• Step 3: Simplify within the square root and resolve the addition or subtraction.
• Step 4: Divide by \( 2a \) to get the roots.

Ultimately, the quadratic formula provides a consistent method for solving any quadratic equation, whether the discriminant \((b^2 - 4ac)\) yields real or complex roots.