Problem 44 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+4 x+5=0 $$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(x^2 + 4x + 5 = 0\) are \(-2 + i\) and \(-2 - i\).

Step 1: Identify the coefficients

In this quadratic equation \(x^2 + 4x + 5 = 0\), the coefficients are \(a = 1\), \(b = 4\), and \(c = 5\).

Step 2: Calculate the discriminant

The discriminant, denoted by \(D\), can be found using the formula \(D = b^2-4ac\). Substituting the coefficients into the formula gives: \(D = (4)^2 - 4*1*5 = 16 - 20 = -4\).

Step 3: Interpret the discriminant

Since the discriminant (\(-4\)) is less than zero, there are no real roots for this quadratic equation. Instead, two complex roots will exist.

Step 4: Compute the Roots

Now applying the quadratic formula to compute the roots, we find the roots are: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{-4 \pm \sqrt{-4}}{2*1} = \frac{-4 \pm \sqrt{4*-1}}{2} = -2 \pm 1i\). Therefore, the roots of the equation are \(-2 + i\) and \(-2 - i\).

Key Concepts

Complex Roots

To understand complex roots, we first need to know what happens when the discriminant of a quadratic equation is negative. A quadratic equation is generally written in the form \(ax^2 + bx + c = 0\). The discriminant, \(D\), helps determine the nature of the roots of this equation. When \(D < 0\), it means we cannot find real numbers as solutions. Instead, the solutions will involve complex numbers. Complex numbers are written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, where \(i^2 = -1\). This imaginary unit allows us to find the square root of negative numbers, something we can't do within the realm of real numbers. For the given equation \(x^2 + 4x + 5 = 0\), the discriminant is \(-4\). Using the quadratic formula, the solutions become complex numbers: \(-2 + i\) and \(-2 - i\). Here \(-2\) is the real part, and \(\pm i\) are purely imaginary parts.

Discriminant

The discriminant is a key element in understanding the roots of a quadratic equation. Given as \(D = b^2 - 4ac\), it determines the nature of the roots without needing to fully solve the equation.

• If \(D > 0\), there are two distinct real roots;
• If \(D = 0\), there is exactly one real root, also known as a repeated or double root;
• If \(D < 0\), the roots are not real and are instead complex conjugates.

In our equation, \(x^2 + 4x + 5 = 0\), the values \(a = 1\), \(b = 4\), and \(c = 5\) lead to a discriminant of \(-4\), indicating the presence of complex roots. The negative value signposts an entrance into the realm of complex numbers.

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a systematic method to find the roots, as given by:\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \]This formula offers two solutions because of the \(\pm\) symbol, accounting for both the addition and subtraction of the square root value. The function of the quadratic formula expands to handle all potential outcomes for the discriminant: real and distinct, real and repeated, or complex roots.For our quadratic equation \(x^2 + 4x + 5 = 0\), the formula results in complex solutions: \(-2 \pm i\). This clearly illustrates how the formula adapts to a negative discriminant by allowing the involvement of imaginary numbers, leading us to discover the solutions \(-2 + i\) and \(-2 - i\). This highlights the versatility of the quadratic formula in handling a variety of situations.