Question

Write the above information as a second degree equation.

Answer 1

We need to translate the given word problem about a rectangle's dimensions and area into a quadratic (second-degree) equation.

Let the length of the shorter side be a variable, say x.
Express the longer side in terms of x.
Use the formula for the area of a rectangle: Area = length × width.

Let the length of the shorter side of the rectangle be x metres.
According to the problem, the other side is 30 metres longer. So, the length of the longer side is (x + 30) metres.
The area of the rectangle is given as 351 square metres.
Using the area formula:
Area = (shorter side) × (longer side) 351 = x × (x + 30) Now, expand the equation:
351 = x² + 30x To write it in the standard form of a second-degree equation (ax² + bx + c = 0), we move all terms to one side:
x² + 30x - 351 = 0 The required second-degree equation is x² + 30x - 351 = 0.

Answer 2

We need to solve the quadratic equation derived in the previous part to find the value of x, which represents the length of the shorter side.

We need to solve the equation x² + 30x - 351 = 0. We can use the quadratic formula:
x = -b ± √(b² - 4ac)2a where a=1, b=30, and c=-351.

Substitute the values of a, b, and c into the quadratic formula:
x = -30 ± √((30)² - 4(1)(-351))2(1) x = -30 ± √(900 + 1404)2 x = -30 ± √(2304)2 To find the square root of 2304, we can notice that 40² = 1600 and 50² = 2500, so the root is between 40 and 50. Since it ends in 4, the root must end in 2 or 8. Let's test 48: 48 × 48 = 2304.
So, √(2304) = 48.
x = (-30 ± 48)/(2) This gives two possible solutions for x:
Solution 1: x = (-30 + 48)/(2) = (18)/(2) = 9
Solution 2: x = (-30 - 48)/(2) = (-78)/(2) = -39
Since x represents the length of a side of a rectangle, it cannot be negative. Therefore, we discard the value x = -39.
The only valid solution is x = 9.

The length of the shorter side is 9 metres.