Problem 56 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate. $$9 x^{2}+18 x+5=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -\frac{1}{3}\) and \(x = -\frac{5}{3}\).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \(9x^2 + 18x + 5 = 0\). It is in the standard form \(ax^2 + bx + c = 0\), where \(a = 9\), \(b = 18\), and \(c = 5\).

Step 2: Determine the Appropriate Method

To solve the quadratic equation, consider using the quadratic formula as it works universally. The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\):\[b^2 - 4ac = 18^2 - 4 \cdot 9 \cdot 5 = 324 - 180 = 144.\]

Step 4: Apply the Quadratic Formula

Use the quadratic formula to find the roots:\[x = \frac{-18 \pm \sqrt{144}}{2 \cdot 9} = \frac{-18 \pm 12}{18}.\]

Step 5: Solve for the Roots

Calculate the two possible values for \(x\):1. For \(x = \frac{-18 + 12}{18} = \frac{-6}{18} = -\frac{1}{3}\).2. For \(x = \frac{-18 - 12}{18} = \frac{-30}{18} = -\frac{5}{3}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \(ax^2 + bx + c = 0\). It is especially useful because it provides an exact solution and works reliably for any standard quadratic equation. The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The symbol \(\pm\) indicates there are two possible values for \(x\), which correspond to two potential solutions or roots of the equation. By plugging the values \(a\), \(b\), and \(c\) into the formula, you can calculate these roots directly. This method is advantageous because you do not have to factor the quadratic equation manually, especially when the equation doesn’t factor easily. The solutions derived from the quadratic formula can either be real or complex numbers, depending on the discriminant's value.

Discriminant

The discriminant is a key part of the quadratic formula, located under the square root sign \(\sqrt{b^2 - 4ac}\). It plays a crucial role in determining the nature of the roots of the quadratic equation:

• If the discriminant is positive, \(b^2 - 4ac > 0\), the equation will have two distinct real roots.
• If the discriminant is zero, \(b^2 - 4ac = 0\), there will be exactly one real root, and the quadratic equation is said to have a "double root."
• If the discriminant is negative, \(b^2 - 4ac < 0\), the equation has two complex roots.

In our example, the discriminant is calculated as \(144\), which is positive. This indicates that the quadratic equation \(9x^2 + 18x + 5 = 0\) has two distinct real roots. Understanding the discriminant helps in predicting the types of solutions without actually solving the equation completely.

Standard Form

Understanding the standard form of a quadratic equation is essential for effectively using the quadratic formula. This form is expressed as \(ax^2 + bx + c = 0\). Here:

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

For any quadratic equation to be analyzed using standard methods or the quadratic formula, it must be in this format. This visibility of coefficients allows you to directly apply mathematical techniques like factoring or using the quadratic formula.
In the example problem \(9x^2 + 18x + 5 = 0\), it is already in standard form, making it straightforward to identify \(a = 9\), \(b = 18\), and \(c = 5\), which are critical for computing the discriminant and using the quadratic formula effectively.