Problem 90 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(x^{2}-4 x+29=0\)

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-4x+29=0\) are \(x = 2 + 5i\) and \(x = 2 - 5i\).

Step 1 : Express the equation in standard form

The quadratic equation is already in standard form: \(ax^{2} + bx + c = 0\). In our case, \(a = 1\), \(b = -4\) and \(c = 29\).

Step 2 : Calculate the discriminant

We would now find the discriminant of the equation, which is given by \(b^{2}-4ac\). Substituting the values we have: \((-4)^{2}-4(1)(29) = 16 - 116 = -100.\)

Step 3 : Evaluate the roots

Substitute \(a\), \(b\), and \(c\) into the quadratic formula to find the solution. The formula gives us \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Substituting the values of \(a\), \(b\) and \(c\), we get \(x = \frac{-(-4) \pm \sqrt{-100}}{2*1} = \frac{4 \pm \sqrt{-100}}{2}\).\nSince the discriminant is negative, we have complex roots. Recalling that \(\sqrt{-1} = i\), we can rewrite \(\sqrt{-100}\) as \(10i\). Therefore, we have \(x = \frac{4 \pm 10i}{2} = 2 \pm 5i\).

Step 4 : Expressing the final solution

The solutions are both complex numbers. Thus, the equation \(x^{2}-4x+29=0\) has two complex solutions: \(x = 2 + 5i\) and \(x = 2 - 5i\).

Key Concepts

Complex Numbers

Complex numbers are essential when dealing with equations that have negative discriminants. A complex number is often represented as a combination of a real part and an imaginary part, typically given in the form \(a + bi\). Here, \(a\) is the real component, and \(b\) is the imaginary component multiplied by the imaginary unit \(i\), where \(i\) represents the square root of \(-1\).

• Imaginary Part: It involves the elusive mathematical construct \(i = \sqrt{-1}\). When a number has a negative inside the square root, the result is imaginary.
• Real Part: The real component, such as \(a\) in \(a + bi\), is simply a conventional number that we encounter in everyday mathematics.

In the context of solving quadratics, when the discriminant (the value under the square root in the quadratic formula) is negative, you will always end up with complex numbers as the solutions. For our original problem, the solutions were \(2 + 5i\) and \(2 - 5i\), demonstrating typical examples of complex numbers with both real and imaginary parts.

Quadratic Formula

The quadratic formula is a powerful tool that solves any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula helps find the roots of the quadratic equation and is especially useful when factoring is not straightforward. The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• The symbol \(\pm\) indicates that there are usually two solutions.
• The values of \(a\), \(b\), and \(c\) are known coefficients from the quadratic equation.
• The numerical expression under the square root, \(b^2 - 4ac\), is called the discriminant and determines the nature of the roots.

In our example problem, substituting \(a = 1\), \(b = -4\), and \(c = 29\) into the quadratic formula helps us calculate the solutions efficiently, revealing their complex nature due to the negative discriminant.

Discriminant

The discriminant is a pivotal element in determining the nature of the roots of a quadratic equation. Denoted by \(D\), the discriminant is calculated using the expression \(b^2 - 4ac\) derived from the coefficients of the equation: \(ax^2 + bx + c = 0\). The discriminant not only reveals the type of solutions but also their number.

• If \(D > 0\): The quadratic equation has two distinct real roots.
• If \(D = 0\): The equation has exactly one real root, or it is said to have a repeated root.
• If \(D < 0\): The equation has two complex roots, which are conjugates of each other.

In the problem given, the discriminant was \(-100\), which is less than zero, confirming that the equation has two complex conjugate solutions. This direct calculation helps predict the nature of the solutions before solving the entire quadratic formula.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is crucial in understanding and solving quadratic problems. It appears as \(ax^2 + bx + c = 0\). Here, \(x\) is the variable, and \(a\), \(b\), and \(c\) are coefficients with specific roles in the equation:

• \(a\): The coefficient of \(x^2\) and must be non-zero. It determines the parabola's "width" or "narrowness" when plotted.
• \(b\): The coefficient of \(x\). It helps in determining the direction the parabola shifts along the x-axis.
• \(c\): The constant term. It affects the vertical shift of the parabola along the y-axis.

In the provided equation \(x^2 - 4x + 29 = 0\), the standard form was correctly recognized. Understanding each coefficient's role is crucial for manipulating and solving the equation using methods such as factoring and applying the quadratic formula.