Problem 42 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{2}+8 x+17=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -4 + i\) and \(x = -4 - i\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 8\), and \(c = 17\).

Step 2: Apply the Quadratic Formula

To solve the quadratic equation \(x^2 + 8x + 17 = 0\), we use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 1\), \(b = 8\), and \(c = 17\) into this formula.

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Here, \(b^2 = 64\) and \(4ac = 68\). Thus, the discriminant is \(64 - 68 = -4\).

Step 4: Evaluate the Roots

Since the discriminant is negative (\(-4\)), the quadratic equation has two complex roots. Use the formula: \[x = \frac{-8 \pm \sqrt{-4}}{2} \].\Since \(\sqrt{-4} = 2i\), the solutions are: \(x = \frac{-8 \pm 2i}{2} = -4 \pm i\).

Step 5: Final Answer

The solutions of the equation \(x^2 + 8x + 17 = 0\) are complex numbers \(x = -4 + i\) and \(x = -4 - i\).

Key Concepts

Complex Solutions

When working with quadratic equations, sometimes you’ll encounter solutions that are not real numbers. Specifically, if the discriminant (more about this later) is negative, the solutions are complex numbers. Complex solutions appear in the form of \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\). In our example, the equation \(x^2 + 8x + 17 = 0\) yields complex roots because the discriminant is negative. This means there is no real solution that will satisfy the equation if you think of it visually, as the parabola does not touch the x-axis. The solutions for our equation turn out to be \(-4 + i\) and \(-4 - i\), illustrating how complex solutions come in conjugate pairs. Conjugate pairs mean that the two solutions are mirror images with respect to the imaginary axis, just swapping the plus for the minus. Understanding this symmetry is important as it ensures the solutions are correctly paired with complex numbers.

Quadratic Formula

The quadratic formula is a vital tool for solving any quadratic equation, typically in the form \(ax^2 + bx + c = 0\). No matter the type of coefficients (integers, fractions, or decimals), this formula efficiently provides solutions. It's expressed as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how you apply it:

• Identify coefficients: \(a\), \(b\), and \(c\).
• Plug these into the formula.
• Compute the discriminant, \(b^2 - 4ac\), right under the square root.
• Solve for \(x\) using both the plus and minus variants in the formula.

In this exercise's equation, \(a = 1\), \(b = 8\), and \(c = 17\). Substituting in, our quadratic formula proceeds to reveal complex roots due to a negative discriminant, which we solve further using imaginary numbers.

Discriminant

The discriminant is the part of the quadratic formula that dictates the nature of the roots you will receive from a quadratic equation. Given by \(b^2 - 4ac\), the discriminant decides whether the roots are real or complex:

• If positive, you get two distinct real roots.
• If zero, one real root exists, repeated (both roots are the same).
• If negative, you end up with two complex roots.

In the case of our equation, the discriminant is calculated as \(64 - 68 = -4\). Since it’s negative, our solutions involve complex numbers. Knowing how to quickly assess the sign of a discriminant helps in predicting the nature of the solutions without solving the entire equation. So, when you see a quadratic equation, find the discriminant to understand what type of solutions to expect: real or complex. This insight saves time and aids in planning your solving strategy effectively.