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One side of a square was shortened by 3 inches and the other side was lengthened by 5 inches. The area of the new square is 55 inches greater than half the area of the original square. Find the length of a side of the original square

User Auralbee
by
7.5k points

2 Answers

2 votes

Answer:

10

Explanation:

Let's say the side of a square is
s

Then the area of the original square would be
s^2

After modification, the sides of the new quadrilateral would be
s-3 and
s+5

Meaning that the area of the new quad. which is equal to
(s^2)/(2) +55 would also be equal to
(s-3)(s+5)

Using the FOIL method, we get


(s-3)(s+5)\\s^2+5s-3s-15\\s^2+2s-15

That means that
(s^2)/(2) +55
=s^2+2s-15

Multiplying both sides by two gets us to
s^2+110=2s^2+4s-30

Subtract both sides by
s^2 --->
110=s^2+4s-30

Subtract
110 from both sides --->
s^2+4s-140=0

Factor --->
(s+14)(s-10)

Since the side of a square cannot be negative, that eliminates the option of
s=-14 and leaves us with an answer of
s=10.

If doubts still remain, we can simply check this by plugging in the number 10:

Original area=
10^2 or
100


10-3=7, 10+5=15


15*7=105


(1)/(2) of the original area is
50\\


50+55=105

Meaning that
10 is the correct answer.

User Steharro
by
8.2k points
7 votes

Answer:

10 inches

Explanation:

Let x be the length of one side of the original square.

According to the question:

One side is shortened by 3 inches can be expressed as (x-3)

Another side was lengthened by 5 inches can be expressed as (x+5)

We know that:


\sf \textsf{The area of the original square }(A_O)=Length*Length = x* x=x^2.


\sf \textsf{The area of the New square }(A_N)=Length*Length =(x-3)(x+5).

Again, we know that

The area of the new square is 55 inches greater than half the area of the original square.

This can be expressed as:


\sf A_N - 55 = (A_O)/(2)

Substituting value


\sf (x-3)(x+5) -55 =(x^2)/(2)

Expanding the left side of the equation, we get


\sf x(x+5)-3(x+5) -55 =(x^2)/(2)


\sf x^2+5x-3x-15 -55 =(x^2)/(2)


\sf x^2+2x - 70 = (x^2)/(2)

doing Criss cross multiplication


\sf 2x^2+4x-140 = x^2

keeping x^2 in left side


\sf 2x^2 - x^2 +4x -140 =0


\sf x^2 +4x -140 = 0

We can solve for x using the quadratic formula:


\sf x = (-b\pm √(b^2 - 4ac))/( 2a)

Comparing above equation with
\sf \underline{ax^2 +bx+ c}

In this case, a = 1, b = 4, and c = -140.

Substituting these values into the quadratic formula, we get:


\sf x = (-4 \pm √(4^2 - 4 * 1 * -140))/( 2 * 1)


\sf x = (-4 \pm √(576))/(2)


\sf x = (-4 \pm 24)/(2)

Taking positive


\sf x = (-4 + 24)/(2)

x = 10

Taking Negative


\sf x = (-4 - 24)/(2)

x = -14

Since length of sqaure is always positive.

Therefore, the length of a side of the original square is 10 inches.

User Gessica
by
8.7k points