A rectangle has an area of (x^2 – 17x + 72) square units. Since the area of a rectangle is determined using the formula, A = lw, what could be the length and width of the rectangle
The Correct Answer and Explanation is:
To find the possible length and width of a rectangle with an area of x2−17x+72x^2 – 17x + 72×2−17x+72 square units, we need to factor the given quadratic expression. The area of a rectangle is given by the formula:A=l×wA = l \times wA=l×w
So, we want to express the area x2−17x+72x^2 – 17x + 72×2−17x+72 as a product of two binomials, which will represent the possible length and width of the rectangle.
Step 1: Factor the quadratic expression
We are given:x2−17x+72x^2 – 17x + 72×2−17x+72
We need two numbers that multiply to 72 and add to -17. These numbers are -9 and -8 because:−9×−8=72and−9+(−8)=−17-9 \times -8 = 72 \quad \text{and} \quad -9 + (-8) = -17−9×−8=72and−9+(−8)=−17
So we factor the expression:x2−17x+72=(x−9)(x−8)x^2 – 17x + 72 = (x – 9)(x – 8)x2−17x+72=(x−9)(x−8)
Step 2: Interpret the factors
Now that we have factored the expression, the area becomes:A=(x−9)(x−8)A = (x – 9)(x – 8)A=(x−9)(x−8)
This means the rectangle could have a length of (x – 9) units and a width of (x – 8) units, or vice versa.
Explanation
Factoring quadratic expressions helps us determine dimensions that, when multiplied, result in the original expression. In this case, since the area of the rectangle is given as x2−17x+72x^2 – 17x + 72×2−17x+72, we look for two binomials whose product equals the area. Factoring reveals that (x−9)(x−8)(x – 9)(x – 8)(x−9)(x−8) is equivalent to x2−17x+72x^2 – 17x + 72×2−17x+72. These two expressions represent potential measurements of the rectangle.
The reason we factor is because the formula for the area of a rectangle is based on multiplication. To reverse that operation and find possible dimensions, we use factoring. Understanding this connection between algebraic expressions and geometric shapes helps bridge abstract and real-world problem solving. This also reinforces the concept that different binomial factors can represent dimensions of a shape while maintaining the same total area.
