Problem 54 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-3-3 x$$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Rearrange the Equation

First, let's rearrange the given equation into a standard quadratic form. The original equation is:\[x^2 = -3 - 3x\]Subtract \(-3 - 3x\) from both sides to set the equation to zero:\[x^2 + 3x + 3 = 0\]

Step 2: Analyze the Quadratic for Real Solutions

Now, we need to determine if the quadratic equation has real solutions. Recall that the quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Identify the coefficients from \(x^2 + 3x + 3 = 0\): \(a = 1\), \(b = 3\), \(c = 3\). Calculate the discriminant \( \Delta = b^2 - 4ac \):\[\Delta = 3^2 - 4 \cdot 1 \cdot 3 = 9 - 12 = -3\]

Step 3: Interpret the Discriminant

The discriminant \(\Delta = -3\) is less than zero, which means the quadratic equation has no real solutions. This is because the square root of a negative number is not a real number.

Step 4: Conclusion

Given that the discriminant is negative, we conclude that the equation \(x^2 + 3x + 3 = 0\) has no real solutions. Thus, it is not possible to represent these solutions on a graph with real numbers.

Key Concepts

Discriminant

In solving quadratic equations, the discriminant is a crucial component. It's a part of the quadratic formula used to determine the nature of solutions. The discriminant is represented by the expression \(b^2 - 4ac\). This value tells us whether the roots of a quadratic equation are real or complex.

Here's what the discriminant can reveal:

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution (also called a repeated or double root).
• If the discriminant is negative, the equation has no real solutions and instead has two complex solutions.

In our example, when we calculated \(b^2 - 4ac\) and found it to be \(-3\), it indicated that there are no real solutions. Instead, the solutions are complex.

Real Solutions

When we talk about real solutions in the context of quadratic equations, we refer to the intersections of the quadratic curve (a parabola) with the x-axis. Equations with real solutions will have points where they actually cross this axis.

To determine if a quadratic equation has real solutions, we rely on the discriminant:

• A positive discriminant means the parabola intersects the x-axis at two points, giving two distinct real solutions.
• A zero discriminant means the parabola just touches the x-axis, leading to one real solution.
• A negative discriminant means the parabola does not touch the x-axis at all, resulting in no real solutions. The solutions are instead complex numbers.

In our step-by-step solution, the calculated discriminant was \(-3\), confirming no x-axis intersections, and therefore, no real solutions for the equation \(x^2 + 3x + 3 = 0\).

Quadratic Formula

The quadratic formula is a tool used to find the solutions to any quadratic equation. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. To solve this equation, we use:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]This formula comprises several elements:

• \(-b\): This part accounts for the linear coefficient.
• \(\pm \sqrt{b^2 - 4ac}\): This part involves the discriminant, deciding the nature of the solutions.
• \(2a\): This is the denominator that scales the solution.

However, as seen in our example, if the discriminant (\(b^2 - 4ac\)) is negative, the square root of this negative number results in complex numbers. This tells us that the solution set is not on the real plane, but rather involves imaginary numbers. Hence, for the equation \(x^2 + 3x + 3 = 0\), there are no real solutions upon applying the quadratic formula.