krissy asked in Science & MathematicsMathematics · 1 decade ago

# Math Question....Factoring?

The sum of the squares of two consecutive positive integers is 145. Find the integers. Please explain step by step...Thanks

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• 1 decade ago

x^2 + (x+1)^2 = 145

x^2 + x^2 + 2x +1 = 145

2x^2 + 2x -144 = 0

(2x - 16)(x + 9) = 0

x = 8 or -9 are your roots, but you want positive integers, so ignore -9.

x = 8 and x+1 = 9 are the two consecutive positive integers.

• 1 decade ago

(8, 9)

First, call the integers x and x+1. Then, square each and add them together: x^2; x^2 + 2x + 1 yields x^2 + x^2 + 2x + 1 = 2x^2 + 2x + 1. Then set that sum equal to 145 and re-arrange to have the standard quadratic form: 2x^2 + 2x + 1 = 145 (subtract 145 from each side) so: 2x^2 + 2x + 1 - 145 = 145 - 145 so: 2x^2 + 2x - 144 = 0. Then substitute into the quadratic equation's answer form with a = 2, b = 2, and c = -144. This yields two answers, 8 and -9. Since the answer requires two positive consecutive integers, we drop the -9. Since we let x equal the smaller of the two integers, the second one is 9. The answer then is (8, 9). Checking, we find 64 + 81 does equal 145.

• Puggy
Lv 7

Let x be one of the two consecutive integers. Being consecutive, it follows that the next one is x + 1.

The sum of their squares is 145; translating that into an algebraic equation,

x^2 + (x + 1)^2 = 145

Solving for x, we get

x^2 + x^2 + 2x + 1 = 145

2x^2 + 2x + 1 = 145

2x^2 + 2x - 144 = 0

Divide by 2,

x^2 + x - 72 = 0

Factor,

(x + 9)(x - 8) = 0

Which implies x = {-9, 8}.

Since we have two solutions for x, it follows that we have two sets of solutions.

The consecutive numbers are -9 and -8, or 8 and 9.

• 1 decade ago

two consecutive integers, x and x+1

sum of squares (x)^2 + (x+1)^2 equals 145

x^2 + x^2 + 2x + 1 = 145

2x^2 + 2x - 144 = 0

(2x + 18)(x - 8) = 0

x = 8 or -9 (but you said positive so...)

the integers are 8 and 9

• Amit Y
Lv 5

Two consecutive positive integers are m and m+1.

Let's find them knowing that

m^2 + (m + 1)^2 = 145

m^2 + m^2 + 2m + 1 = 145

2m^2 + 2m = 144

2m^2 + 2m - 144 = 0

m^2 + m - 72 - 0

m1,2 = (-1 +- sqrt(1 + 288))/2 = (-1 +- sqrt(289))/2 =

= (-1 +- 17)/2

m1 = 16/2 = 8

m2 = -18/2 = -9 -> Not a solution because it is negative

8^2 + 9^2 = 64 + 81 = 145

• John T
Lv 6

So, you have two positive integers, n and n+1, since they are consecutive.

n^2 + (n+1)^2 = 145

n^2 + n^2 + 2n + 1 = 145

2n^2 + 2n - 144 = 0

n^2 +n -72 = 0 (multiply through by 1/2 )

(n+9) (n-8)

answers are both -9 and 8. There are two solutions.