Problem 64 - Elementary Algebra

Question

Solve. $$ (x-9)(2 x+3)=2(x-9) $$

Step-by-Step Solution

Answer

The solutions are \(x = 9\) and \(x = -\frac{1}{2}\).

Step 1: Expand the Equation

First, let's expand the left-hand side of the equation \((x-9)(2x+3)\). Use the distributive property: \(a(b+c) = ab + ac\).If we expand, we get: \[(x-9)(2x+3) = x(2x+3) - 9(2x+3) = 2x^2 + 3x - 18x - 27\]Combine the like terms: \[2x^2 - 15x - 27\]

Step 2: Set the Equation Equal to Zero

Now that we have expanded the left-hand side, we'll set the equation equal to zero. Rewrite the original equation with the expanded left side:\[2x^2 - 15x - 27 = 2(x-9)\]Distribute the right side:\[2x^2 - 15x - 27 = 2x - 18\]Subtract \(2x - 18\) from both sides to set the equation to zero:\[2x^2 - 15x - 2x - 27 + 18 = 0\]Combine the like terms again:\[2x^2 - 17x - 9 = 0\]

Step 3: Factor the Quadratic Equation

Next, we factor the quadratic equation \(2x^2 - 17x - 9 = 0\). We need two numbers that multiply to \(2 imes (-9) = -18\) and add to \(-17\). Those numbers are \(-18\) and \(1\).Rewrite the middle term:\[2x^2 - 18x + x - 9 = 0\]Factor by grouping:\[(2x^2 - 18x) + (x - 9) = 0\]\[2x(x - 9) + 1(x - 9) = 0\]Factor out the common factor \((x - 9)\):\[(x - 9)(2x + 1) = 0\]

Step 4: Solve for x

Now, solve for \(x\) by setting each factor equal to zero:1. \(x - 9 = 0\): \[x = 9\]2. \(2x + 1 = 0\): \[2x = -1\] \[x = -\frac{1}{2}\]

Step 5: Verify the Solutions

It is always a good habit to double-check solutions in the original equation. Substitute \(x = 9\) into the original equation:Left side: \[(9-9)(2\times9+3) = 0\]Right side: \[2(9-9) = 0\]This is true.Now, substitute \(x = -\frac{1}{2}\):Left side: \[\left(-\frac{1}{2} - 9\right)(2\times -\frac{1}{2} + 3) = \left(-\frac{19}{2}\right)\left(2\right) = 2\left(-\frac{19}{2}\right) = -19\]Right side: \[2\left(-\frac{1}{2} - 9\right) = -19\]This is true.

Key Concepts

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic equation in a way that turns it into a product of simpler expressions. This makes solving the equation much easier. In the example, the quadratic equation we focused on was \(2x^2 - 17x - 9 = 0\). The trick here is to find two numbers that multiply to the product of the leading coefficient and the constant term, which is \(-18\) in this case, and at the same time, add up to the middle coefficient, which is \(-17\). These numbers are \(-18\) and \(1\). To break down these steps:

• Identify the product: multiply the leading coefficient \(2\) by the constant \(-9\), resulting in \(-18\).
• Identify a sum: find two numbers that add up to \(-17\), the coefficient of \(x\).
• Rewrite \(-17x\) as \(-18x + x\), splitting it according to the numbers discovered.
• Group and factor the expression step by step: use common factoring on the grouped terms to eventually isolate the factors \((x-9)(2x+1)\).

This turns the problem of solving a quadratic equation into finding the roots of simple linear expressions. It's a straightforward yet powerful method!

Distributive Property

The distributive property is a fundamental concept in algebra, allowing us to simplify expressions and equations. It states that if you have an expression in the form \(a(b + c)\), you can expand it as \(ab + ac\). In our original exercise, it helps to expand the left side of the equation \((x-9)(2x+3)\).Here's how it works:

• First, multiply \(x\) with each term inside the second bracket: \(x \cdot 2x\) and \(x \cdot 3\), resulting in \(2x^2 + 3x\).
• Then, apply the same rule to \(-9\), so \((-9) \cdot 2x\) and \((-9) \cdot 3\), which gives us \(-18x - 27\).

By applying the distributive property, this expression becomes a sum of terms: \(2x^2 + 3x - 18x - 27\). It simplifies the comparison of different sides in equations, a crucial step in both solving and verifying equations.

Solving Equations

Solving quadratic equations effectively finds the values of \(x\) that make the equation true. With the factored form of the equation \((x-9)(2x+1) = 0\), you can easily find these solutions. Here's how to solve the equation step by step:

• Use the zero-product property: if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Hence, set each factor equal to zero.
• For \(x-9 = 0\), solving gives \(x = 9\).
• For \(2x+1 = 0\), solve by first isolating \(x\):
• Subtract \(1\) from both sides: \(2x = -1\).
• Divide by \(2\) to find \(x\): \(x = -\frac{1}{2}\).

Solving these linear expressions provides the solutions \(x = 9\) and \(x = -\frac{1}{2}\), uncovering the points where the quadratic turns to zero. This process is essential to understanding how equations reflect on graphs and in real-world problems.

Algebraic Verification

Algebraic verification is the process of ensuring that our solutions satisfy the original equation. It's an essential part of solving equations to confirm both accuracy and thoroughness. In this problem, after finding \(x = 9\) and \(x = -\frac{1}{2}\), we checked each solution in the original equation.Let's verify the solutions:

• Substitute \(x = 9\) back into the original equation: \((9-9)(2\times9+3) = 2(9-9)\).
• Both sides simplify to \(0\), confirming \(x = 9\) is correct.
• For \(x = -\frac{1}{2}\), substitute into the equation: substitute to find that both sides equal \(-19\), so this is also accurate.

These steps ensure that solutions work for the initial equation and verify the problem is solved correctly. Verification not only confirms that solutions are correct, but also helps to understand the consistency of algebraic principles.