Problem 58 - Intermediate Algebra

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 complex: \(x = -3 \pm i\sqrt{2}\).

Step 1: Rewrite the Equation in Standard Form

First, let's rewrite the equation in the standard quadratic form by moving all terms to one side. Start with the given: \[ x^2 + 6x = -11 \] Add 11 to both sides:\[ x^2 + 6x + 11 = 0 \] Now, the equation is in the form \( ax^2 + bx + c = 0 \).

Step 2: Identify the Coefficients

In this quadratic equation, identify the coefficients: \( a = 1 \), \( b = 6 \), and \( c = 11 \). These will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) will help us solve the quadratic equation. Substitute \( a = 1 \), \( b = 6 \), and \( c = 11 \) into the formula:\[x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}\]

Step 4: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \) to determine if the roots are real or complex numbers:\[6^2 - 4 \cdot 1 \cdot 11 = 36 - 44 = -8\]The discriminant is \(-8\), which is less than zero, indicating that the roots are complex.

Step 5: Solve for Complex Roots

Since the discriminant is negative, use the quadratic formula to find the complex roots:\[x = \frac{-6 \pm \sqrt{-8}}{2}\]Simplify the square root of \(-8\):\[\sqrt{-8} = \sqrt{-1 \cdot 4 \cdot 2} = 2i\sqrt{2}\]Thus, the solutions are:\[x = \frac{-6 \pm 2i\sqrt{2}}{2}\]Which simplifies further to:\[x = -3 \pm i\sqrt{2}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in algebra for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). It provides a straightforward solution by calculating the roots of the equation. The formula is given by:

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

Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation. "\( b^2 - 4ac \)" within the formula is known as the discriminant, and it plays a crucial role in determining the nature of the roots. Using this formula allows us to find the exact solutions, whether they are real or complex. It's particularly useful for cases where simple factoring isn't possible, or when the roots aren't integers.

Complex Roots

Complex roots occur in quadratic equations when the discriminant \( b^2 - 4ac \) is negative. In such cases, the equation cannot be solved by finding real numbers because taking the square root of a negative number isn't possible within the real number system.

• This is where complex numbers come in handy. A complex number can be expressed as \( a + bi \), where \( i \) is the imaginary unit with the property that \( i^2 = -1 \).
• When using the quadratic formula and the discriminant is negative, the square root of the negative discriminant becomes an imaginary number. For instance, \( \sqrt{-8} \) simplifies to \( 2i\sqrt{2} \).
• In our example, the solution results in complex roots of the form \( x = -3 \pm i\sqrt{2} \), indicating that the solutions are not crossing the x-axis on the graph.

This means the solutions, or roots, have no real part crossing into the real number plain but exist entirely in the realm of complex values.

Discriminant

The discriminant is a mathematical expression found in the quadratic formula under the square root sign: \( b^2 - 4ac \). Its primary role is to determine the nature and number of roots for the quadratic equation. It can tell you at a glance whether the roots are real or complex.

• If the discriminant is positive \((b^2 - 4ac > 0)\), the quadratic equation has two distinct real roots.
• If the discriminant is zero \((b^2 - 4ac = 0)\), there is exactly one real root, which is repeated.
• If the discriminant is negative \((b^2 - 4ac < 0)\), the roots of the equation will be complex, meaning they have a real part and an imaginary part.

In our specific problem, the discriminant came out as \(-8\), meaning the roots were complex as calculated further in the steps.

Standard Form

In quadratic equations, the standard form is an essential way of writing the equation. It is expressed as \( ax^2 + bx + c = 0 \), where:

• \( a \), \( b \), and \( c \) are numerical coefficients.
• \( x \) is the variable, and it signifies square terms \( x^2 \) as the highest power.

Rewriting an equation in standard form is the first crucial step to solving it using the quadratic formula because it organizes all terms on one side of the equation. This consistent form allows us to easily identify the coefficients \( a \), \( b \), and \( c \) needed for further calculations, like finding the discriminant or applying the quadratic formula. In the problem, moving terms to get \( x^2 + 6x + 11 = 0 \) illustrates the transformation into standard form.