Respuesta :
Answer:
width = 4ft
length = 13
Step-by-step explanation:
Let the unknown value of width be y; and the unknown value of length be x
4 times the width = 4y
The length, x is 3 feet less than 4y
= [tex]4y - 3[/tex]
A board is rectangular in shape.
The formula for calculating the area of a rectangle is;
Area= width x length
Area of the bulletin board is given as 52 ft²
Width = y
Length = 4y - 3
Therefore,
[tex]52=y(4y-3)[/tex]
[tex]52=4y²-3y[/tex]
[tex]4y²-3y-52=0[/tex]
This leads to a quadratic equation which has the form [tex]ax²+bx+c=0[/tex]
Steps to solving a quadratic equation are
1. Multiply the coefficient of x² by c, i.e, a times c = [tex]ac[/tex]
2. We look for two factors of ac that can also be added together to give b.
[tex]4y²-3y-52=0[/tex] corresponds to [tex]ax²+bx+c=0[/tex]
4 x -52 = -208
Two factors of -208 that can add up to give -3 are -16 and 13
We can write [tex]4y²-3y-52=0[/tex] as [tex](4y²-16y)+(13y-52)=0[/tex]
We factorise each brackets
[tex]4y(y-4)+13(y-4)=0[/tex]
This becomes,
[tex](4y+13)(y-4)=0[/tex]
[tex]4y+13=0[/tex]
[tex]y-4=0[/tex]
[tex]4y=-13[/tex]
[tex]4y/4=-13/4[/tex]
[tex]y=-13/4[/tex]
Or
[tex]y-4=0[/tex]
[tex]y=4[/tex]
A quadratic equation has two solutions. Therefore, [tex]y = -13/4[/tex]or [tex]y=4[/tex]
But the width of a rectangle is usually a natural number hence, the possible value of the width y, is = 4
Length, [tex]x=4y-3[/tex]
but y = 4
Therefore length, x=[tex]4(4)-3=16-3=13[/tex]
