Problem 61 - Precalculus : Building Concepts and Connections

Question

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$x^{2}+2 x+3=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2}+2 x+3=0\) are complex and given by \(x=-1 \pm i\sqrt{2}\). These solutions are also the zeros of the related quadratic function \(f(x) = x^{2}+2 x+3\).

Step 1: Understanding the Quadratic Equation

The given equation is \(x^{2}+2 x+3=0\). A quadratic equation is of the form \(ax^{2}+bx+c=0\) where a, b and c are coefficients.

Step 2: Identify the Coefficients

Looking at the given quadratic equation, identify the coefficients as follows: \(a = 1\), \(b = 2\), \(c = 3\).

Step 3: Apply the Quadratic Formula

The quadratic formula which gives the solutions or roots of a quadratic equation is \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Substituting the values of a, b and c, we get \(x=\frac{-2\pm\sqrt{(2^2)-(4*1*3)}}{2*1}\).

Step 4: Simplify the Expression

Calculate the value under the square root (the discriminant): \(b^2-4ac=(2^2)-(4*1*3)=4-12=-8\). Then simplify the entire expression to obtain \(x=\frac{-2\pm \sqrt{-8}}{2}\).

Step 5: Derive the Roots

Given \(x=\frac{-2\pm \sqrt{-8}}{2}\), the values in the square root indicates that solutions are complex. Complex roots occur in conjugate pairs. As a result we have \(x=\frac{-2\pm 2i\sqrt{2}}{2}\) derived from \(\sqrt{-8}\). Simplifying gives \(x=-1 \pm i\sqrt{2}\).

Step 6: Relate to Zeros of Quadratic Function

The quadratic function related to the given equation is \(f(x) = x^{2}+2 x+3\). The zeros of the function are the values of x where the function equals zero. These are the solutions to the equation \(x^{2}+2 x+3=0\) which as calculated are \(x=-1 \pm i\sqrt{2}\).

Key Concepts

Complex Roots

In mathematics, the concept of complex roots arises when solving quadratic equations that do not intersect the x-axis. This happens when the discriminant (the value under the square root in the quadratic formula) is negative. Complex roots indicate that the solutions to the quadratic equation involve imaginary numbers. Typically, complex roots appear in pairs known as conjugates. For instance, if a quadratic equation simplifies to have a root of the form \(-1 + i ext{√}2\), its pair would be \(-1 - i ext{√}2\). These roots are crucial in understanding equations with no real solutions, broadening our exploration of solutions to include the complex plane.

• Complex numbers involve an imaginary unit denoted as i, where i = √-1.
• Conjugate roots help maintain the real nature of polynomial coefficients.
• Complex roots mean the graph of the function never touches the x-axis.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions, or roots, of a quadratic equation. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).To solve for the variable \(x\), we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].This formula allows us to calculate the roots of any quadratic equation, regardless of the nature of the roots (real or complex).

• The formula requires identifying \(a\), \(b\), and \(c\), which are the coefficients from the equation.
• It combines basic algebra with the concept of square roots.
• The sign \(\pm\) reflects the potential for two solutions, owing to the parabola's possible intersection at two points on the x-axis.

By substituting the values of these coefficients into the formula, we can determine the roots efficiently. This method is reliable and comprehensive for diverse quadratic scenarios.

Discriminant

The discriminant is a significant part of the quadratic formula, represented by \(b^2 - 4ac\).It determines the nature of the roots of a quadratic equation. This value can tell us whether the roots are real or complex, and whether they are distinct or repeated.

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots, which is the case for our example \(x^2+2x+3=0\) where the discriminant is -8.

Understanding the discriminant is vital for predicting the number and type of solutions without fully solving the equation. It provides a quick check of what to expect in terms of roots, assisting in both comprehending and graphing quadratic functions.

Zeros of a Function

The zeros of a function, also known as roots or solutions, are the values of \(x\) that make the function equal to zero.For a quadratic function \(f(x) = ax^2 + bx + c\), the zeros are the solutions to the equation \(ax^2 + bx + c = 0\).In the context of the quadratic equation \(x^2 + 2x + 3 = 0\), the zeros are complex: \(-1 \pm i\text{√}2\).

• The zeros showcase the points where the graph of a quadratic function would intersect the x-axis if they exist as real numbers.
• Complex zeros suggest there are no x-intercepts, and the graph sits above or below the x-axis.
• Identifying zeros is crucial in graphing and understanding functions, as they help define the shape and position of the parabola.

Since the zeros relate directly to the equation's solutions, mastering this concept helps in analyzing quadratic equations and predicting their behaviors on a graph.