Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Solve the equation. $$(2 x+9)(4 x-3)=8 x^{2}-12$$

Step-by-Step Solution

Answer

The solution is \(x = \frac{1}{2}\).

Step 1: Expand the Left Side

Start by expanding the left side of the equation: \((2x + 9)(4x - 3)\). Use the distributive property: first distribute \(2x\), then distribute \(+9\). This gives:\[2x(4x) + 2x(-3) + 9(4x) + 9(-3) = 8x^2 - 6x + 36x - 27\]Simplify by combining like terms:\[8x^2 + 30x - 27\]

Step 2: Equate Both Sides of the Equation

After expanding and simplifying the left hand side, equate it to the right side of the equation given:\[8x^2 + 30x - 27 = 8x^2 - 12 \]

Step 3: Eliminate the Quadratic Terms

Subtract \(8x^2\) from both sides. This will eliminate the quadratic terms:\[30x - 27 = -12 \]

Step 4: Solve for x

Add 27 to both sides to isolate the term with \(x\):\[30x - 27 + 27 = -12 + 27 \]This simplifies to:\[30x = 15\]Now divide both sides by 30 to solve for \(x\): \[x = \frac{15}{30} = \frac{1}{2}\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, often used to simplify equations. This property states that multiplying a number by a group of numbers added together is the same as doing the multiplication individually. For example, in an expression like \((a + b)(c + d)\), each term inside the first set of parentheses should be multiplied by each term in the second set of parentheses.

This is expressed as

• \(a(c + d) + b(c + d)\), or
• \(ac + ad + bc + bd\).

The distributive property makes it easy to simplify and solve equations by breaking down complex expressions into manageable parts. In the context of the original exercise, we used the distributive property to expand \((2x + 9)(4x - 3)\). This step was essential for simplifying the equation and making further computation possible.

Solving Linear Equations

Linear equations are equations of the first order. This means they involve no exponents higher than one, forming a straight line when plotted on a graph. Solving linear equations involves finding the value of the unknown variable that makes the equation true.

Key steps in solving linear equations include:

• Isolating terms involving the variable on one side of the equation.
• Ensuring that each step maintains the equality by performing the same operation on both sides.
• Solving for the variable to identify its value.

In our exercise, the equation was simplified to \(30x - 27 = -12\). By systematic manipulation, we isolated \(x\) to solve for it. Adding 27 to both sides ensured balance, leading to \(30x = 15\), and dividing both sides by 30 provided the solution \(x = \frac{1}{2}\).

Expanding Binomials

Expanding binomials involves applying the distributive property to multiply two binomials. A binomial is a polynomial with two terms, like \((a + b)\) or \((c - d)\).

The purpose of expanding binomials is to simplify an expression or form equations that are easier to work with. When expanding, each term of one binomial is multiplied by each term of the other. An expanded binomial typically results in a polynomial with four terms, which can often be simplified by combining like terms.

For instance, in the exercise at hand, expanding the binomial \((2x + 9)(4x - 3)\) resulted in four terms:

• \(8x^2\), from \(2x \times 4x\),
• \(-6x\), from \(2x \times -3\),
• \(36x\), from \(9 \times 4x\), and
• \(-27\), from \(9 \times -3\).

These were combined to yield the simplified expression \(8x^2 + 30x - 27\), facilitating progress on the equation.