Problem 27 - Algebra Form and Function

Question

For what values of \(x\) are the quantities negative? $$ x^{2}+x-2 $$

Step-by-Step Solution

Answer

Short Answer: The quadratic expression \(x^2+x-2\) is negative for values of \(x\) in the interval \(-2 < x < 1\).

Step 1: Find the roots of the quadratic expression

To find the roots of the expression \(x^2+x-2\), we will solve the equation \(x^2+x-2=0\). This is a quadratic equation of the form \(ax^2+bx+c=0\), where \(a=1\), \(b=1\), and \(c=-2\). To solve it, we can use the quadratic formula: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Applying the quadratic formula, we get: $$ x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-2)}}{2(1)} $$ $$ x = \frac{-1 \pm \sqrt{9}}{2} $$ $$ x = \frac{-1 \pm 3}{2} $$ Therefore, we have two roots, \(x_1 = 1\) and \(x_2 = -2\).

Step 2: Create intervals of \(x\) values

Now that we have the roots, we can create intervals of \(x\) values: 1. \(x<-2\) 2. \(-21\)

Step 3: Determine the sign of the quadratic expression for each interval

We can determine the sign of the quadratic expression for each interval by substituting a value from each interval into the expression \(x^2+x-2\) and checking the result's sign. For interval \(x<-2\), let's choose a value, say \(x=-3\): $$ (-3)^2 + (-3) - 2 = 9 - 3 - 2 = 4 > 0 $$ The expression is positive in this interval. For interval \(-21\), let's choose a value, say \(x=2\): $$ 2^2 + 2 - 2 = 4 + 2 - 2 = 4 > 0 $$ The expression is positive in this interval.

Step 4: Conclusion

The quadratic expression \(x^2+x-2\) is negative for values of \(x\) in the interval \(-2 < x < 1\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that allows us to find the roots, or solutions, of any quadratic equation in the standard form \( ax^2 + bx + c = 0 \).
This formula is expressed as:

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

It solves for \( x \) by dealing with the three coefficients: \( a \), \( b \), and \( c \).
In the context of the exercise, our equation is \( x^2 + x - 2 = 0 \), where \( a = 1 \), \( b = 1 \), and \( c = -2 \).
Plugging these into the quadratic formula allows us to find the specific values of \( x \) that make this equation equal to zero.
It’s important to note how the plus-minus symbol (\( \pm \)) in the formula divides it into two different solutions, which is why we typically find two roots from a quadratic equation.

Roots of Quadratic Equation

Finding the roots of a quadratic equation means determining the values of \( x \) where the equation equals zero. In simpler terms, roots are the x-values where the equation touches or crosses the x-axis in a graph.
For our specific equation, \( x^2 + x - 2 = 0 \), we already used the quadratic formula to compute these values.
Here's a recap:

• The roots we found were \( x_1 = 1 \) and \( x_2 = -2 \).

These values are critical because they divide the number line into intervals, helpful in determining other properties of the equation, such as when the quadratic is positive or negative.

Sign of Expression Intervals

Understanding the sign of a quadratic expression within specific intervals is crucial for resolving inequalities and analyzing quadratic behavior over a range of \( x \)-values.
In this exercise, after determining the roots \( x_1 = 1 \) and \( x_2 = -2 \), our number line is split into three parts:

• \( x < -2 \)
• \( -2 < x < 1 \)
• \( x > 1 \)

To assess the expression's sign in each interval, substitute a test value from each section back into the equation \( x^2 + x - 2 \).

• For \( x < -2 \) (e.g., \( x = -3 \)), the expression is positive.
• For \( -2 < x < 1 \) (e.g., \( x = 0 \)), the expression is negative.
• For \( x > 1 \) (e.g., \( x = 2 \)), the expression is positive.

In conclusion, the quadratic expression is negative within the interval \( -2 < x < 1 \).
This kind of interval analysis helps us understand and predict where the function is increasing, decreasing, or remaining positive or negative.