Problem 58 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$x^{2}+6 x=-11$$

Step-by-Step Solution

Answer

The solutions are \(-3 + i\sqrt{2}\) and \(-3 - i\sqrt{2}\).

Step 1: Rearrange the Equation

First, let's rearrange the given equation to the standard quadratic form. The original equation is \(x^2 + 6x = -11\). By moving the \(-11\) to the left side, we get:\[ x^2 + 6x + 11 = 0 \]

Step 2: Determine the Method of Solution

The quadratic equation \(x^2 + 6x + 11 = 0\) can be solved using the quadratic formula, since it does not factor easily. The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 1\), \(b = 6\), and \(c = 11\).

Step 3: Calculate the Discriminant

The discriminant \(D\) of the quadratic equation is calculated as:\[ D = b^2 - 4ac \]Substitute the values to calculate \(D\):\[ D = 6^2 - 4(1)(11) = 36 - 44 = -8 \]

Step 4: Apply the Quadratic Formula

Since the discriminant \(D\) is negative, the solutions will be complex numbers. Substitute the values into the quadratic formula:\[ x = \frac{-6 \pm \sqrt{-8}}{2(1)} \]This simplifies to:\[ x = \frac{-6 \pm 2i\sqrt{2}}{2} \]Divide each term by 2:\[ x = -3 \pm i\sqrt{2} \]

Step 5: Identify the Solution

The solutions to the quadratic equation \(x^2 + 6x + 11 = 0\) are the complex numbers:\[ x = -3 + i\sqrt{2} \quad \text{and} \quad x = -3 - i\sqrt{2} \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions for quadratic equations. Quadratic equations are of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The formula helps us find the roots, or solutions, of these equations. The quadratic formula is:

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

Using this formula, we can solve any quadratic equation, even if it doesn't factor nicely. It allows us to find both real and complex solutions, depending on the value of the discriminant \( D \).

Complex Solutions

Quadratic equations can sometimes have complex solutions. These solutions occur when the discriminant is negative. Complex numbers are of the form \( a + bi \), where \( i \) is the imaginary unit with the property \( i^2 = -1 \). In our example:

• The discriminant \( D = b^2 - 4ac \) is \(-8\).
• Because \( D \) is negative, the solutions involve imaginary numbers.

This means that instead of intersecting the x-axis, the parabola represented by the equation doesn't touch or cross the x-axis on a graph. Complex solutions often show up in scenarios where physical phenomena, oscillations, or alternating current (AC) electricity are described.

Discriminant

The discriminant is a component of the quadratic formula that helps determine the nature of the solutions. It's given by the expression \( D = b^2 - 4ac \). The value of the discriminant tells us several things:

• If \( D > 0 \), the equation has two distinct real roots.
• If \( D = 0 \), the equation has one real root, meaning the parabola touches the x-axis at one point.
• If \( D < 0 \), the equation has two complex roots, indicating that the solutions include imaginary numbers.

In solving the equation \( x^2 + 6x + 11 = 0 \), we found \( D = -8 \), which led us to expect complex solutions.

Standard Quadratic Form

The standard quadratic form is crucial in solving quadratic equations. A quadratic equation must be expressed in the form \( ax^2 + bx + c = 0 \) to use the quadratic formula and other algebraic methods effectively. This process often involves rearranging the given equation by moving all terms to one side of the equation:

• Our starting equation \( x^2 + 6x = -11 \) was rearranged to \( x^2 + 6x + 11 = 0 \).

This standard form makes it straightforward to identify the coefficients \( a \), \( b \), and \( c \) needed for applying the quadratic formula. Transitioning to standard quadratic form is an essential step in solving these types of equations.