Problem 27 - Introductory Algebra for College Students

Question

Solve each quadratic equation using the quadratic formula. $$x^{2}-6 x+13=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 3+2i\) and \(x = 3-2i\)

Step 1: Identify \(a\), \(b\), and \(c\)

From the equation \(x^{2}-6 x+13=0\) we can see that \(a = 1\), \(b = -6\), and \(c = 13\)

Step 2: Substitute the values into the quadratic formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula \(x= {[-b±sqrt(b^2-4ac)]/(2a)}\). This gives us \(x = [(6±sqrt((-6)^{2}-4*1*13)]/(2*1)}\)

Step 3: Calculate the discriminant

Calculate the value of the discriminant \(b^2 - 4ac\) = \((-6)^2 - 4*1*13 = 36 - 52 = -16\) which is negative. Hence, the roots of the equation are complex.

Step 4: Calculate the value of \(x\)

Substitute discriminant into the formula to get the roots. This gives us \(x = [6±sqrt(-16)]/2 = [6±4i]/2 = 3±2i\)

Key Concepts

complex roots

When we talk about complex roots, we refer to solutions of an equation that involve imaginary numbers. Complex roots often arise when the discriminant of a quadratic equation is negative. A complex number includes a real part and an imaginary part. For instance, if we have a solution like \( 3 \pm 2i \), then 3 is the real component, and \( 2i \) represents the imaginary part.

• An imaginary number is expressed in terms of \( i \), where \( i = \sqrt{-1} \).
• If the solutions of an equation include \( i \), then they are considered complex.

Understanding complex roots is crucial as they frequently occur in various mathematical scenarios, especially when solving quadratic equations. Not all quadratic equations have complex roots, but when they do, it's often due to a negative discriminant. This means that the quadratic equation graphs do not intersect the x-axis, indicating that there are no real roots.

discriminant

The discriminant in a quadratic equation is a critical component that helps determine the nature of the roots without having to solve the entire equation. The formula for the discriminant is \( b^2 - 4ac \). Depending on its value, the discriminant signals whether an equation has two real roots, one real root, or two complex roots.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is one real root, sometimes referred to as a repeated or double root.
• If the discriminant is negative, the equation has two complex roots.

In our example, the discriminant calculation of \( (-6)^2 - 4 \times 1 \times 13 \) resulted in a negative value of \(-16\). This directly leads to complex roots, indicating no intersection with the x-axis on a graph of the equation.

quadratic equation

A quadratic equation is a polynomial equation of the second degree, typically in the form of \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). Quadratic equations are crucial as they appear in many areas of mathematics and problem solving. Their solutions can be found using different methods, such as factoring, completing the square, or using the quadratic formula.

• The standard form is \( ax^2 + bx + c = 0 \).
• The graph of a quadratic equation is a parabola.
• Each quadratic equation has either none, one, or two real roots, or complex roots based on the discriminant.

Quadratic equations are often introduced in these terms to build a solid foundation for understanding a range of mathematical concepts that extend beyond basic algebra.

solving equations

Solving equations is a fundamental skill in mathematics involving finding the values of variables that satisfy the equation. For quadratic equations, solutions can be obtained using the quadratic formula when other methods like factoring are not feasible. The quadratic formula is a universal solution method, expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula uses the coefficients \( a \), \( b \), and \( c \) directly from the quadratic equation.

• Identify coefficients \( a \), \( b \), and \( c \) from the equation \( ax^2 + bx + c = 0 \).
• Calculate the discriminant, \( b^2 - 4ac \).
• Use the quadratic formula to find solutions.

By following these steps, problems that seem complicated at first glance can often be broken down into manageable parts. The use of the quadratic formula provides an effective way to achieve solutions, highlighting the importance of understanding discriminants and roots.