Problem 70 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$(3 x+4)(3 x-1)-33 x=-20$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{4}{3}\).

Step 1: Expand the Expression

Start by expanding the left-hand side of the equation \[(3x + 4)(3x - 1) = 3x \cdot 3x + 3x \cdot (-1) + 4 \cdot 3x + 4 \cdot (-1) \]Simplify the terms:\[9x^2 - 3x + 12x - 4\]Combine like terms:\[9x^2 + 9x - 4\]The equation becomes\[9x^2 + 9x - 4 - 33x = -20\]

Step 2: Simplify the Equation

Combine like terms by subtracting \(33x\) from \(9x\):\[9x^2 - 24x - 4 = -20\] To isolate \(0\) on one side, add \(20\) to both sides:\[9x^2 - 24x + 16 = 0\]

Step 3: Identify the Coefficients

In the standard form of a quadratic equation, \(ax^2 + bx + c = 0\), identify the coefficients:\[a = 9, \ b = -24, \ c = 16\]

Step 4: Apply the Quadratic Formula

Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Plug the coefficients into the formula:\[x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 \cdot 9 \cdot 16}}{2 \cdot 9}\]Compute the discriminant:\[(-24)^2 - 4 \cdot 9 \cdot 16 = 576 - 576 = 0\]

Step 5: Solve for x

Since the discriminant is 0, there is one real solution:\[x = \frac{24 \pm \sqrt{0}}{18}\]Simplify:\[x = \frac{24}{18} = \frac{4}{3}\]

Key Concepts

Expanding Expressions

Expanding expressions involves removing parentheses and writing the expression in a longer form. This is usually done by distributing each term in one of the parentheses to the terms in the other parentheses. Consider the expression \[(3x + 4)(3x - 1)\].To expand this, you must multiply every term in the first set of brackets by every term in the second set of brackets. This process is sometimes referred to as the FOIL method, which stands for First, Outer, Inner, Last, representing the terms you multiply together. Here’s how it looks:

• First: Multiply the first terms: \(3x \cdot 3x = 9x^2\)
• Outer: Multiply the outer terms: \(3x \cdot (-1) = -3x\)
• Inner: Multiply the inner terms: \(4 \cdot 3x = 12x\)
• Last: Multiply the last terms: \(4 \cdot (-1) = -4\)

So, after expanding, the expression becomes \[9x^2 - 3x + 12x - 4\].This expansion helps in later steps like combining like terms.

Combining Like Terms

Once you have expanded an expression, the next step is to simplify it by combining like terms. Like terms are terms that have identical variable parts. For instance, in the expression \[9x^2 - 3x + 12x - 4\],\(-3x\) and \(12x\) are like terms because they both contain the same variable raised to the same power.To combine these terms, you perform the arithmetic operation indicated by the signs in front of them. So, \(-3x + 12x\) results in \(9x\).Now the expression looks like \[9x^2 + 9x - 4\].Combining like terms simplifies the equation, making it easier to solve especially in equations leading to quadratic expressions.

Simplifying Equations

Simplifying an equation involves reducing it to its simplest form, where it is easier to apply algebraic methods. Once you’ve combined like terms, the next goal is to isolate terms on one side of the equation, usually starting by moving constants around. Consider the equation \[9x^2 + 9x - 4 - 33x = -20\].Here, we combine \(9x\) and \(-33x\) to get \(-24x\).This gives us \[9x^2 - 24x - 4 = -20\].Next, we want to bring the constant term from the right side over to the left side. To do this, add \(20\) to both sides, resulting in \[9x^2 - 24x + 16 = 0\].The equation is now in its simplest form, ready for solving using methods like the quadratic formula.

Solving Quadratic Equations

Solving quadratic equations typically involves finding the value(s) of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). One powerful method for finding these values is the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].For the equation \[9x^2 - 24x + 16 = 0\],identify \(a = 9\), \(b = -24\), and \(c = 16\).Input these into the quadratic formula:\[x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 \cdot 9 \cdot 16}}{2 \cdot 9}\].Calculate the discriminant \((-24)^2 - 4 \cdot 9 \cdot 16\). Here, it equals 0, indicating a single real solution.Solving the equation \[x = \frac{24 \pm \sqrt{0}}{18}\] simplifies to \[x = \frac{24}{18} = \frac{4}{3}\].Using the quadratic formula provides solutions even when manual factoring is complex or impossible. Here, it gives a clear path to finding the solution.