The area of a rectangle is \( 2 x^{2}-11 x+15 \) square units. What are the dimensions of the length and width?

To find the dimensions of the rectangle, we first need to factor the expression for the area, \( 2x^{2} - 11x + 15 \). We can start by looking for two numbers that multiply to \( 2 \times 15 = 30 \) and add up to \( -11 \). The numbers \( -5 \) and \( -6 \) do the trick! We can rewrite the quadratic as \( 2x^{2} - 5x - 6x + 15 \) and then factor by grouping. This gives us: \[ (x - 3)(2x - 5) \] Thus, the dimensions of the rectangle are \( x - 3 \) units (width) and \( 2x - 5 \) units (length). To ensure you get the right values, double-check that both expressions are positive for the dimensions to make sense. For instance, setting \( x - 3 > 0 \) gives \( x > 3 \) and ensuring \( 2x - 5 > 0 \) gives \( x > 2.5 \). Hence, just make sure \( x \) is larger than 3 for valid rectangle dimensions!