The area of a rectangle is given by the formula: $\text{Area} = \text{Base} \times \text{Height}.$

Given:

• Base = $2x + 10$
• Height = $3x$
• Area = 288 square inches

Substitute the given values into the formula: $288 = (2x + 10) \times (3x).$

Expand the right-hand side: $288 = 6x^2 + 30x.$

Now, rearrange this into a standard quadratic equation: $6x^2 + 30x - 288 = 0.$

To simplify this equation, divide all terms by 6: $x^2 + 5x - 48 = 0.$

Now, solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},$ where $a = 1$, $b = 5$, and $c = -48$.

Substitute these values into the formula: $x = \frac{-5 \pm \sqrt{5^2 - 4 \times 1 \times (-48)}}{2 \times 1},$ $x = \frac{-5 \pm \sqrt{25 + 192}}{2},$ $x = \frac{-5 \pm \sqrt{217}}{2}.$

Calculate the square root of 217 and simplify: $x = \frac{-5 \pm 14.73}{2}.$

This gives two possible values for $x$: $x = \frac{-5 + 14.73}{2} = \frac{9.73}{2} \approx 4.87,$ $x = \frac{-5 - 14.73}{2} = \frac{-19.73}{2} \approx -9.87.$

Since $x$ represents a length, it must be positive. Thus, the value of $x$ is approximately: $x \approx 4.87.$

Would you like more details or have any questions about this solution?

Here are 5 related questions for further exploration:

1. What is the perimeter of the rectangle with the found value of $x$?
2. What are the dimensions of the rectangle when $x \approx 4.87$?
3. How would the problem change if the area of the rectangle were different?
4. What would the quadratic equation look like if the base and height expressions were swapped?
5. How does the quadratic formula help in solving problems like this?

Tip: Always check the feasibility of solutions in real-world contexts, like ignoring negative values for dimensions.