a standard form of a quadratic equation has a form of:
y = ax^2 + bx + c
a is the coefficient of the x^2 term.
b is the coefficient of the x term.
c is the constant term.
the graph of the quadratic equation looks like an umbrella.
it's called a parabola.
if a is positive, the parabola points down and opens up.
if a is negative, the parabola points up and opens down.
the first graph below is the parabola pointing down because the coefficient of the x^2 term is positive.
the second graph below is the parabola pointing up because the coefficient of the x^2 term is negative.
the equations used are:
pointing down / opening up:
y = 2x^2 + 7x - 2
pointing up / opening down:
y = -2x^2 + 7x - 2
a parabola has one maximum point or one minimum point.
if the parabola points down then it has a minimum point.
if the parabola points up then it has a maximum point.
how does this apply to your problem?
see below:
the perimeter of your rectangle is equal to 350 cm.
if we let L = length of the rectangle and W = width of the rectangle, then:
perimeter equals 2L + 2W
area equals L * W.
we know that the perimeter 350 cm.
this means that 2L + 2W = 350
in this equation, we can solve for W to get:
W = (350-2L)/2 which can be simplified to:
W = 275 - L
if we substitute for W in the equation for the perimeter of the rectangle, then we get:
2L + 2*(175-L) = 350
if we let x = L, then this equation becomes:
2x + 2*(175-x) = 350.
since we know that the area of the rectangle equals L * W, we can substitute for W in this equation as well to get:
L * (175-L) = Area of Rectangle.
if we substitute x for L, then we get:
x * (175-x) = Area of Rectangle.
if we let y = area of rectangle, then our equation becomes:
y = x * (175-x)
we simplify this equation to get:
y = 175*x - x^2
this is a quadratic equation.
we can graph this equation to get the diagram below:
you can see that this graph will peak at somewhere around 7500.
we can solve for the max/min point of the equation by using the formula of:
x = -b/2a
our equation is:
y = 175*x - x^2 which we can re-write as:
y = -x^2 + 175*x
this is the standard form of the quadratic equation where:
a = coefficient of the x^2 term is equal to -1.
b = coefficient of x term is equal to 175.
our formula for the max/min point is:
x = -b/2a which comes out to be:
x = -175 / (2*(-1)) which comes out to be:
x = 87.5
when x = 87.5, the value of y becomes:
y = -(87.5)^2 + 175*(87.5) which comes out to be:
y = -7656.25 + 15312.5 which comes out to be:
y = 7656.25
since y represents the area of the rectangle, then the maximum area of the rectangle is equal to 7656.25 square cm.
you can create a table of values to see that this is accurate and that the perimeter will always be 350 cm.
this table will look as follows:
length width perimeter area
x 175-x 2x + 2(175-x) x*(175-x)
45 130 350 5850
85 90 350 7650
87.5 87.5 350 7656.25 ***** (max)
90 85 350 7650
125 50 350 6250
165 10 350 1650
this quadratic equation had a maximum value because the coefficient of the x^2 term was negative.