Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation. $$9 x^{2}-6 x+35=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(9x^{2}-6x+35=0\) are complex. They are \(x = 1/3 ± sqrt(306)*i/6\).

Step 1: Identify a, b and c

In the given quadratic equation \(9x^{2}-6x+35=0\), \(a=9\), \(b=-6\) and \(c=35\). We insert these values into the quadratic formula.

Step 2: Substitute a, b, and c into the quadratic formula

Substituting \(a=9\), \(b=-6\) and \(c=35\) into the quadratic formula, we get \(x = [-(-6) ± sqrt((-6)^2-4*9*35)]/2*9\).

Step 3: Simplify the expression

After substituting the numbers, the next step is simplifying the expression inside the square root and the division. First, simplify the discriminant (the part inside the square root): \(b^2-4ac = (-6)^2-4*9*35 = 36 - 1260 = -1224\). Since, this results in a negative number, the roots are complex.

Step 4: Find the roots of the quadratic equation

After simplifying, the two roots of the quadratic equation are given by \(x = [6 ± sqrt(-1224)]/18 \). It can simplified with \(x = 6/18 ± sqrt(-1224)/18 \), which simplifies to \(x = 1/3 ± sqrt(306)*i/6\). Note that 'i' is the imaginary unit.

Key Concepts

Complex Numbers

Complex numbers are a fascinating concept in mathematics. When the solutions to equations don't nicely fall into positive or negative real numbers, complex numbers come into play. A complex number has two parts: a real part and an imaginary part. The imaginary part involves the square root of a negative number, which is represented by the symbol 'i'.

• The imaginary unit 'i' is defined as the square root of -1, which means that \(i^2 = -1\).
• A complex number is typically written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
• In the context of quadratic equations, complex numbers are especially useful when the discriminant (which we will discuss later) is negative, leading to imaginary roots.

Understanding complex numbers allows us to solve equations that have no real solutions, broadening the scope of equations we can handle. For the equation \(9x^{2}-6x+35=0\), the complex solution was found due to the negative discriminant, indicating a need for 'i'.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows us to find the solutions without needing to factor the expression. The formula is given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here's how the formula works step-by-step:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Plug these values into the formula to compute the roots.
• The symbol '±' indicates that there are generally two solutions: one using plus and one using minus.
• The term under the square root, \(b^2 - 4ac\), is known as the discriminant, which determines the nature of the roots.

The quadratic formula provides an easy pathway to finding solutions, and knowing how to use it is essential for solving any quadratic equation efficiently.

Discriminant in Quadratic Equations

The discriminant of a quadratic equation is a crucial element that helps determine the nature of the roots. It is the part of the quadratic formula under the square root, given by \(b^2 - 4ac\).

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, meaning the quadratic has a perfect square.
• If the discriminant is negative, as in the case of our quadratic equation \(9x^{2}-6x+35=0\), it results in two complex roots.

The discriminant tells us how the parabola represented by the quadratic formula will intersect the x-axis. For a negative discriminant, the parabola does not intersect the x-axis in the real number plane. Instead, the solutions are complex and sit in the imaginary plane. This knowledge aids in correctly anticipating the kind of answers a quadratic equation will yield before diving into calculations.