Question

Solve each equation, if possible. $$ (x+2)(x-3)=(x+3)^{2} $$

Short answer

x = -\frac{15}{7}

Step-by-step solution

- Expand Both Sides

First, expand both sides of the equation. The left side, \( (x+2)(x-3) \), expands to \( x^2 - x - 6 \). The right side, \( (x+3)^2 \), expands to \( x^2 + 6x + 9 \). The equation now looks like this: \[ x^2 - x - 6 = x^2 + 6x + 9 \].

- Eliminate Common Terms

Subtract \( x^2 \) from both sides to eliminate the \( x^2 \) term: \[ -x - 6 = 6x + 9 \].

- Combine Like Terms

Add \ x \ to both sides to combine like terms: \[ -6 = 7x + 9 \].

- Solve for x

Subtract 9 from both sides to isolate terms involving \ x \: \[ -15 = 7x \]. Divide both sides by 7 to solve for \ x \: \[ x = -\frac{15}{7} \].

Key concepts

Expand Expressions

To start solving the quadratic equation, we need to expand both sides of the equation. Expanding expressions means we rewrite them from a product of terms to a sum or difference of terms. For instance, we have \( (x+2)(x-3) \) on the left side and \( (x+3)^2 \) on the right side.

Let's break these down:

• For the left side \( (x+2)(x-3) \), distribute each term in the first parenthesis to each term in the second parenthesis:

\[ (x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6 \]

• For the right side \((x+3)^2\), apply the square of a binomial formula \( (a+b)^2 = a^2 + 2ab + b^2 \):

\[ (x+3)^2 = x^2 + 6x + 9 \] Now, the equation becomes \[ x^2 - x - 6 = x^2 + 6x + 9 \]

This step helps to prepare the equation to eliminate common terms and eventually solve for the variable.

Eliminate Common Terms

Once the expressions are expanded, the next step is to eliminate common terms to simplify the equation. In this case, both sides of the equation have the term \(x^2\).

To remove these terms, subtract \(x^2\) from both sides:

\[ x^2 - x - 6 - x^2 = x^2 + 6x + 9 - x^2 \] Simplifying this gives:

\[ -x - 6 = 6x + 9 \]

By eliminating the common \(x^2\) terms, the equation looks much simpler, and we can focus on combining like terms next.

Combine Like Terms

After eliminating common terms, we need to combine like terms to further condense the equation. In our simplified equation \[ -x - 6 = 6x + 9 \], we notice terms involving x and constant terms.

The goal is to isolate the variable x on one side:

• Add \(x\) to both sides of the equation

\[ -x + x - 6 = 6x + x + 9 \] This reduces to:

\[ -6 = 7x + 9 \]

Now, we have successfully combined like terms, positioning ourselves to solve for x in the final step.

Solve for Variable

The last step is to solve for the variable x. We now have the equation: \[ -6 = 7x + 9 \]

Our goal is to isolate x. To do this, first, subtract 9 from both sides:

• \[ -6 - 9 = 7x + 9 - 9 \]

This simplifies to:

\[ -15 = 7x \]

Next, divide both sides by 7 to isolate x:

• \[ x = -\frac{15}{7} \]

So, the solution to the equation \((x+2)(x-3) = (x+3)^2\) is \[ x = -\frac{15}{7} \]

This approach ensures every part of the problem is tackled clearly to derive the right answer.”