Question

topic: solve quadratic equations by factoring
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
a rectangular vegetable garden will have a width that is 2 feet less than the length, and an area of 48 square feet.
if x represents the length, then the length can be found by solving the equation:
x(x - 2) = 48
what is the length, x, of the garden?
the length is
feet.
the solution is

Explanation:

Step1: Expand the equation

Given the equation \( x(x - 2)=48 \), we first expand the left - hand side. Using the distributive property \( a(b - c)=ab - ac \), here \( a = x \), \( b=x \), \( c = 2 \), so we get \( x^{2}-2x=48 \).

Step2: Rewrite in standard quadratic form

Subtract 48 from both sides of the equation \( x^{2}-2x = 48 \) to get the quadratic equation in standard form \( ax^{2}+bx + c = 0 \). So \( x^{2}-2x-48=0 \).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \( c=- 48 \) and add up to \( b=-2 \). The numbers are - 8 and 6 because \( (-8)\times6=-48 \) and \( -8 + 6=-2 \). So we can factor the quadratic equation as \( (x - 8)(x+6)=0 \).

Step4: Solve for x

Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \). So we set each factor equal to zero:

• If \( x - 8=0 \), then \( x = 8 \).
• If \( x + 6=0 \), then \( x=-6 \).

Since the length of a rectangle cannot be negative, we reject the solution \( x=-6 \).

Answer:

8