Problem 62 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation. $$9 x^{2}-6 x+37=0$$

Step-by-Step Solution

Answer

The given quadratic equation has no real roots as the discriminant is negative.

Step 1: Identify a, b and c

The quadratic equation is in the form \(ax^2 + bx + c = 0\). Comparing this with the given equation \(9x^2 - 6x + 37 = 0\), we have \(a = 9\), \(b = -6\) and \(c = 37\).

Step 2: Apply the quadratic formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We have\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 9 \cdot 37}}{2 \cdot 9}\]

Step 3: Simplify the equation

Simplify the equation above to find its roots. However, we notice that the discriminant \(b^2 - 4ac = (-6)^2 - 4 \cdot 9 \cdot 37\) is negative, meaning the equation has no real roots.

Key Concepts

Understanding the Discriminant

When solving quadratic equations of the form \(ax^2 + bx + c = 0\), the discriminant plays a crucial role. The discriminant is the part of the quadratic formula under the radical sign \(b^2 - 4ac\). It helps determine the nature of the roots of the equation.

Here's what you need to know about the discriminant:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also known as a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation has no real roots, but instead has two complex solutions.

In our example equation \(9x^2 - 6x + 37 = 0\), the discriminant is \((-6)^2 - 4 \cdot 9 \cdot 37 = 36 - 1332 = -1296\). Since this value is negative, we know the equation doesn't intersect the x-axis and has complex solutions.

The Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is a reliable method for finding the roots of any quadratic equation. It is derived from the process of completing the square and provides a solution to quadratic equations even when factoring is difficult or impossible.

Using the quadratic formula requires only the coefficients \(a\), \(b\), and \(c\) from your quadratic equation. Simply plug them into the formula:

• \(b\) is the coefficient of the linear term (\(x\))
• \(a\) is the coefficient of the quadratic term (\(x^2\))
• \(c\) is the constant term

After substituting the coefficients into the formula, you calculate the discriminant (\(b^2 - 4ac\)), which determines the type of solutions you will obtain. For the equation \(9x^2 - 6x + 37 = 0\), substituting into the quadratic formula reveals that we will have complex solutions due to the negative discriminant.

Complex Solutions Explained

Complex solutions arise when the discriminant of a quadratic equation is negative. A complex number has a real part and an imaginary part, written in the form \(a + bi\), where \(i\) is the imaginary unit defined as \(\sqrt{-1}\).

When using the quadratic formula, if \(b^2 - 4ac < 0\), you will end up with a square root of a negative number, which signifies complex numbers in the solutions. To express these solutions, you'll need to work with imaginary numbers:

• The real part will be \(-b/(2a)\)
• The imaginary part will be \(\pm \sqrt{|b^2 - 4ac|}/(2a)i\)

For our equation \(9x^2 - 6x + 37 = 0\), the roots are complex because the discriminant \(-1296\) leads to solutions involving imaginary numbers. Thus, the roots are expressed as \(\frac{6}{18} \pm \sqrt{1296}/18i\), or simplified further.