Problem 108 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(6 z^{2}-9 z+1=0\)

Step-by-Step Solution

Answer

\(z = \frac{9 \pm \sqrt{57}}{12}\)

Step 1 - Identify coefficients

Identify the coefficients in the quadratic equation in the form \(ax^2 + bx + c = 0\). Here, \(a = 6\), \(b = -9\), and \(c = 1\).

Step 2 - Write down the Quadratic Formula

The Quadratic Formula is given by: \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Step 3 - Substitute the coefficients into the formula

Substitute \(a = 6\), \(b = -9\), and \(c = 1\) into the Quadratic Formula: \[z = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(6)(1)}}{2(6)}\]

Step 4 - Simplify the expression inside the square root

Calculate the value inside the square root: \[(-9)^2 - 4(6)(1) = 81 - 24 = 57\]

Step 5 - Continue simplifying

Substitute back into the formula and simplify: \[z = \frac{9 \pm \sqrt{57}}{12}\]

Step 6 - Write final solutions

The two solutions are: \[z = \frac{9 + \sqrt{57}}{12}\] and \[z = \frac{9 - \sqrt{57}}{12}\]

Key Concepts

Coefficients Identification

When solving a quadratic equation using the quadratic formula, the first step is to identify the coefficients. A quadratic equation is typically of the form: \(ax^2 + bx + c = 0\).

Here's how to identify them:

• The coefficient \(a\) is the number in front of \(z^2\). In our equation \(6z^2 - 9z + 1 = 0\), \(a = 6\).
• The coefficient \(b\) is the number in front of \(z\). Here, \(b = -9\).
• The constant coefficient \(c\) is the term with no variable. In our case, \(c = 1\).

Identifying these coefficients correctly is crucial because you will need them to use the quadratic formula properly. Check your work carefully to make sure you have the right values.

Quadratic Equation

A quadratic equation is one of the most common types of polynomial equations. It is called 'quadratic' because 'quad' means square, which involves the variable squared (\(x^2\)). Quadratic equations can represent various real-world scenarios, like projectile motion and area problems.
To solve a quadratic equation using the quadratic formula, the equation must first be in standard form:
\(ax^2 + bx + c = 0\).
If your equation is not in this form, you may need to rearrange it first. Here, the given equation is:\(6z^2 - 9z + 1 = 0\), which is already in the standard form. This ensures that the coefficients we identified earlier (\(a = 6\), \(b = -9\), and \(c = 1\)) are readily applicable.

Solution of Quadratic Equations

Solving a quadratic equation involves finding the values of the variable (\(z\)) that satisfy the equation. One of the most reliable methods for this is the Quadratic Formula: \[z = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\].
Let's apply this formula step-by-step:

• Substitute the coefficients \(a = 6\), \(b = -9\), and \(c = 1\) into the formula.
• Calculate inside the square root: \(b^2 - 4ac\), which for us is \((-9)^2 - 4(6)(1) = 81 - 24 = 57\).
• Write the expression under the square root back in the quadratic formula: \[z = \frac{9 \pm \sqrt{57}}{12}\].
• Simplify to find the two possible values of \(z\): \[z = \frac{9 + \sqrt{57}}{12}\] and \[z = \frac{9 - \sqrt{57}}{12}\].

In summary, solving quadratic equations with the quadratic formula involves substituting the coefficients, performing operations inside the square root, and simplifying the results to find the solutions for \(z\). Proper understanding and practice will make this process second nature.