Problem 61 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 9 a(a-3)=3 a-25 $$

Step-by-Step Solution

Answer

The solution is \(a = \frac{5}{3}\).

Step 1 - Expand the Left Side

Start with the left side of the equation: \(9a(a - 3)\). Use the distributive property to expand this to \(9a^2 - 27a\).

Step 2 - Set the Equation to 0

Write the expanded equation as \(9a^2 - 27a = 3a - 25\). To prepare for solving, move all terms to one side by subtracting \(3a\) and adding 25 from both sides, resulting in \(9a^2 - 30a + 25 = 0\).

Step 3 - Solve the Quadratic Equation

Recognize this as a quadratic equation of the form \(ax^2 + bx + c = 0\). Using the quadratic formula \(a = 9\), \(b = -30\), \(c = 25\), substitute into the formula: \[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Calculate the discriminant: \(b^2 - 4ac = (-30)^2 - 4 \times 9 \times 25 = 900 - 900 = 0\). Substitute into the formula: \[ a = \frac{30 \pm \sqrt{0}}{18} = \frac{30}{18} = \frac{5}{3} \].

Step 4 - Verify the Solution

Substitute \(a = \frac{5}{3}\) back into the original equation to verify: \[9 \left(\frac{5}{3}\right)\left(\frac{5}{3} - 3\right) = 3 \left(\frac{5}{3}\right) - 25\]. Calculate each side: The left side simplifies to: \[9 \times \frac{5}{3} \times \left(-\frac{4}{3}\right) = -20\]. The right side simplifies to: \[5 - 25 = -20\]. Both sides are equal, confirming \(a = \frac{5}{3}\) is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental property of arithmetic and algebra that lets you simplify expressions when you multiply a single term by multiple terms in parentheses. In the context of our problem, this property allows us to expand the expression \(9a(a-3)\) into \(9a^2 - 27a\).

Here's how it works step-by-step for this specific case:

• Multiply \(9a\) by each term inside the parenthesis: \(a\) and \(-3\).
• First, \(9a \times a = 9a^2\).
• Next, \(9a \times -3 = -27a\).

Putting it all together, you get \(9a^2 - 27a\). Breaking down an expression like this is crucial because it sets up the equation into a standard quadratic form, making it easier to solve later.

Quadratic Formula

The quadratic formula is your go-to tool for solving quadratic equations, which are of the form \(ax^2 + bx + c = 0\). Once we have our equation in this standard form, applying the quadratic formula becomes straightforward:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s how it applies to our problem:

• First, identify the coefficients \(a = 9\), \(b = -30\), and \(c = 25\).
• We substitute these into the formula to determine the solutions for \(x\).
• In our example, the discriminant, \(b^2 - 4ac\), equals 0. This simplifies the formula since \(\sqrt{0} = 0\).

Because of the discriminant being zero, there's only one solution: \(x = \frac{-b}{2a} = \frac{30}{18} = \frac{5}{3}\). This shows us that using this method, we can find the precise solution for quadratic equations efficiently.

Discriminant in Quadratic Equations

The discriminant in a quadratic equation gives us valuable information about the nature and number of the solutions. It is represented as the expression \(b^2 - 4ac\) within the quadratic formula.

Let's explore what different values of the discriminant indicate:

• If the discriminant is positive, there are two distinct real solutions.
• If it's zero, there's exactly one real solution. This is the case in our problem where \(b^2 - 4ac = 900 - 900 = 0\).
• If the discriminant is negative, no real solutions exist, but two complex solutions will.

Understanding the discriminant is crucial as it quickly informs you whether to expect one, two, or no real solutions without fully solving the equation. It can greatly facilitate the problem-solving process by guiding your expectations and calculations.