Problem 39 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

There are no real solutions for the equation.

Step 1: Rearrange the equation

Start by rearranging the given equation to set it equal to zero. The equation is given as \(2x^2 + 2x = -1\). Add 1 to both sides to isolate all terms on one side: \[2x^2 + 2x + 1 = 0\] Now the equation is in standard quadratic form.

Step 2: Identify coefficients

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). For the equation \(2x^2 + 2x + 1 = 0\), identify the coefficients: \(a = 2\), \(b = 2\), \(c = 1\). These are needed for the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substitute the coefficients \(a = 2\), \(b = 2\), and \(c = 1\) into the formula: \[x = \frac{-2 \pm \sqrt{2^2 - 4(2)(1)}}{2(2)}\] Simplify the terms inside the square root: \(x = \frac{-2 \pm \sqrt{4 - 8}}{4}\)

Step 4: Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots.\(b^2 - 4ac = 4 - 8 = -4\).Since the discriminant is negative, this equation has complex roots, indicating there are no real solutions.

Step 5: Conclude from the discriminant

The negative discriminant \(-4\) tells us that there are no real solutions for this quadratic equation. Thus, there is no need to graph the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It works universally for any quadratic equation, making it a reliable method when other methods like factoring are not feasible. The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here's a quick rundown on how to use it:

• Identify the coefficients \(a\), \(b\), and \(c\) in the quadratic equation.
• Substitute these values into the quadratic formula.
• Solve for \(x\) by performing the arithmetic inside the square root (radical) first, followed by the division and addition/subtraction as indicated.

This formula helps not only in finding real roots but also indicates the nature and type of roots through the part under the square root, known as the discriminant. The quadratic formula is essential because it gives insight into the solutions without needing to graph the equation.

Discriminant

The discriminant is a key component of the quadratic formula, defined as the expression \( b^2 - 4ac \). It plays a critical role in determining the nature of the roots of a quadratic equation. Depending on its value, you can predict whether the roots are real or complex:

• If the discriminant is positive, \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If the discriminant is zero, \( b^2 - 4ac = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, \( b^2 - 4ac < 0 \), the equation has two complex roots, which means there are no intersections with the x-axis on the graph.

In our solved problem, the discriminant calculated as \(-4\) indicates the presence of complex roots due to its negative value. This tells us directly that there aren’t any real numbers that satisfy the equation.

Complex Roots

Complex roots come into play when the discriminant of a quadratic equation is negative. Instead of crossing or touching the x-axis as real roots do, complex roots are represented in the form \(a \pm bi\), where \(i\) is the imaginary unit (\(i^2 = -1\)). Essentially, they denote pairs of roots that are not visible on a standard number line or graph.When solving a quadratic equation using the quadratic formula, if the value under the square root is negative, you will end up with complex roots. For example:\[x = \frac{-b \pm \sqrt{-4}}{2a} = \frac{-b \pm 2i}{2a}\]In such cases, the expression simplifies to yield two roots in a format involving the imaginary unit, indicating an integral characteristic of quadratic equations beyond real numbers. Understanding complex roots is important as it highlights the full spectrum of solutions a quadratic equation can have, providing a complete picture of its behavior.