Problem 42 - Elementary Algebra

Question

Solve. $$x x 2-9=1 x-3$$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -2\).

Step 1: Distribute and Simplify

We start by expanding and distributing any terms necessary on both sides of the equation. However, it seems there is a typo with the equation given. Let's assume the equation is incorrectly typed and correctly assumed to be \(x^2 - 9 = x - 3\).

Step 2: Move all terms to one side

Next, move all terms in the equation to one side to set the equation to zero. This involves subtracting \(x\) and adding 3 to both sides:\[x^2 - 9 - x + 3 = 0\]This simplifies to:\[x^2 - x - 6 = 0\]

Step 3: Factor the quadratic equation

We factor the quadratic equation \(x^2 - x - 6 = 0\). We look for two numbers that multiply to -6 and add to -1. These numbers are -3 and 2:\[(x - 3)(x + 2) = 0\]

Step 4: Solve for x

Set each factor equal to zero and solve for \(x\):1. \(x - 3 = 0\) gives \(x = 3\)2. \(x + 2 = 0\) gives \(x = -2\)

Step 5: Verify solutions

Substitute each solution back into the original equation to verify:1. For \(x = 3\), \(x^2 - 9 = 3 - 3\) gives \(9 - 9 = 0\), true.2. For \(x = -2\), \((-2)^2 - 9 = -2 - 3\) gives \(4 - 9 = -5\), true.Both solutions are verified.

Key Concepts

Factoring Quadratics

Factoring quadratics is a key technique when dealing with quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The idea is to express the equation as the product of two binomials. This can make it much easier to find the solutions to the equation.When you're factoring a quadratic such as \(x^2 - x - 6 = 0\), you look for two numbers that not only add up to the coefficient of \(x\) (which in this case is -1) but also multiply to the constant term (in this case -6). For our example, those numbers are -3 and 2. This is because:

• -3 + 2 = -1 (the middle term coefficient)
• -3 * 2 = -6 (the constant term)

So, the equation can be rewritten as \((x - 3)(x + 2) = 0\). Factoring is a strategy that turns a complex equation into simpler pieces, easing the path to solving.

Solving Equations

Once a quadratic equation is factored into the form \((x - 3)(x + 2) = 0\), solving it becomes straightforward. The principle here is to use the zero product property. According to this property, if the product of two numbers is zero, then at least one of the terms must be zero.This means you set each factor to zero and solve these simple linear equations:

• For \(x - 3 = 0\), add 3 to both sides and you find \(x = 3\).
• For \(x + 2 = 0\), subtract 2 from both sides and you get \(x = -2\).

Each solution comes from assuming one of the factors is zero, given that their product equals zero.

Verifying Solutions

Verifying solutions is the all-important step to ensure that the solutions we derived are correct. It involves substituting the solutions back into the original equation to see if the equation holds true.For our example:

• Substitute \(x = 3\) back into the original equation \(x^2 - 9 = x - 3\):
• Calculating gives \(3^2 - 9 = 3 - 3\), which simplifies to \(0 = 0\).
• This is true, so \(x = 3\) is indeed a solution.
• Next, substitute \(x = -2\) into \(x^2 - 9 = x - 3\):
• Calculate and find that \((-2)^2 - 9 = -2 - 3\), or \(4 - 9 = -5\).
• This equation also holds true, confirming \(x = -2\) as a valid solution.

By substituting back and confirming the truth of the equation, we verify that our solutions are indeed correct and applicable to the original equation.