Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}+2 x=-1$$

Step-by-Step Solution

Answer

No real solutions, as the discriminant is negative.

Step 1: Convert to Standard Form

We need the quadratic equation in the standard form, which is \(ax^2 + bx + c = 0\). Start with the original equation \(2x^2 + 2x = -1\). To convert it, add \(1\) to both sides. This gives us \(2x^2 + 2x + 1 = 0\).

Step 2: Identify Coefficients

In the equation \(2x^2 + 2x + 1 = 0\), identify the coefficients: \(a = 2\), \(b = 2\), and \(c = 1\). This will help in solving it using the quadratic formula.

Step 3: Use the Quadratic Formula

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

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute the values: \(2^2 - 4 \times 2 \times 1 = 4 - 8 = -4\). Since the discriminant is negative, there are no real solutions to the equation.

Step 5: Conclusion on Real Solutions

Since the discriminant is negative, the quadratic equation \(2x^2 + 2x + 1 = 0\) has no real solutions. Graphically, this suggests that the parabola does not intersect the x-axis.

Key Concepts

Understanding the Standard Form of a Quadratic Equation

A quadratic equation involves terms up to the second degree and is typically expressed in its standard form as \(ax^2 + bx + c = 0\). Here:

• \(a\), \(b\), and \(c\) are constants with \(a eq 0\), ensuring the presence of an \(x^2\) term.
• \(x\) represents the variable or unknown to solve for.

In the original equation \(2x^2 + 2x = -1\), our first step is to bring it to standard form. We do this by adding \(1\) to both sides, resulting in \(2x^2 + 2x + 1 = 0\). This rearrangement is crucial for further analysis, as it sets the stage for easy identification of coefficients (\(a = 2\), \(b = 2\), and \(c = 1\)). These coefficients will be used in applying the quadratic formula, which requires the equation to be in this standard format.

Exploring the Discriminant

The discriminant is a significant element within the quadratic formula, symbolized by \(b^2 - 4ac\). It plays a critical role in determining the nature of the solutions of a quadratic equation. By examining the value of the discriminant, we can predict:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution (roots are repeated).
• If negative, the equation has no real solutions, but rather complex ones.

In our exercise, the discriminant is calculated as \(2^2 - 4 \, \times \, 2 \, \times \, 1 = 4 - 8 = -4\). Because this value is negative, it confirms that the equation \(2x^2 + 2x + 1 = 0\) does not possess real solutions. This insight also informs us that the graphical representation of this equation would show a parabola that does not meet the x-axis.

Deciphering the Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a powerful tool that allows us to find the solutions for any quadratic equation in standard form. Here, the coefficients \(a\), \(b\), and \(c\) are plugged into the formula, and the solutions for \(x\) are derived:

• The "\(\pm\)" sign indicates there may be two possible solutions: one adding the square root term and the other subtracting it.
• Prior calculation of the discriminant \(b^2 - 4ac\) is crucial, as it impacts the nature of the resulting solutions.

For our equation \(2x^2 + 2x + 1 = 0\), substituting the values \(a = 2\), \(b = 2\), and \(c = 1\) into the formula, we notice the negative discriminant \(-4\) leads under the square root, confirming all solutions are imaginary or non-real. Hence, this explains why no real intersection points occur on the x-axis.

Real Solutions and Their Implications

Real solutions to a quadratic equation are values of \(x\) that satisfy the equation and can be plotted on a real number line:

• When a quadratic has real solutions, the graph, a parabola, intersects the x-axis.
• If there are no real solutions, as seen when the discriminant is negative, the parabola lies entirely above or below the x-axis without touching it.
• An understanding of real solutions helps compare and interpret different scenarios of parabolic motion in physics, economics, and natural phenomena.

In the exercise provided, the equation \(2x^2 + 2x + 1 = 0\) has no real solutions, as indicated by its negative discriminant \(-4\). Consequently, the graph of this equation is a parabola opening upwards, which entirely floats either above or below the x-axis, without intersecting it. This graphical reflection reinforces our algebraic find that real solutions do not exist for this quadratic equation.