SOLUTION: The larger of two numbers is 5 more than twice the smaller. The square of the smaller is equal to the larger. Find the numbers

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Question 390980: The larger of two numbers is 5 more than twice the smaller. The square of the smaller is equal to the larger. Find the numbers
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

let larger number be y and smaller number x
if the larger of is {{5}}} more than twice the smaller number, we have:
y+=+2x+%2B+5 ..........(1)
if the square of the smaller is equal to the larger,we have
y+=+x%5E2.........(2)
solve the system:
y+=+2x+%2B+5 ..........(1)
y+=+x%5E2.........(2).......substitute in (1)
-----------------------------

x%5E2=+2x+%2B+5 ..........(1).solve for x

x%5E2+-5+=+2x+
x%5E2+-2x++-5+=+0+


Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2-2%2Ax-5=0 ( notice a=1, b=-2, and c=-5)





x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A-5+%29%29%2F%282%2A1%29 Plug in a=1, b=-2, and c=-5




x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A-5+%29%29%2F%282%2A1%29 Negate -2 to get 2




x+=+%282+%2B-+sqrt%28+4-4%2A1%2A-5+%29%29%2F%282%2A1%29 Square -2 to get 4 (note: remember when you square -2, you must square the negative as well. This is because %28-2%29%5E2=-2%2A-2=4.)




x+=+%282+%2B-+sqrt%28+4%2B20+%29%29%2F%282%2A1%29 Multiply -4%2A-5%2A1 to get 20




x+=+%282+%2B-+sqrt%28+24+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%282+%2B-+2%2Asqrt%286%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%282+%2B-+2%2Asqrt%286%29%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%282+%2B+2%2Asqrt%286%29%29%2F2 or x+=+%282+-+2%2Asqrt%286%29%29%2F2



Now break up the fraction



x=%2B2%2F2%2B2%2Asqrt%286%29%2F2 or x=%2B2%2F2-2%2Asqrt%286%29%2F2



Simplify



x=1%2Bsqrt%286%29 or x=1-sqrt%286%29



So the solutions are:

x=1%2Bsqrt%286%29 or x=1-sqrt%286%29






x1=+1+%2Bsqrt%286%29=1%2B2.45
x1=+3.45..................................
x2=+1+%2Bsqrt%286%29=1-2.45
x2=+-1.45...................................
find y
y1+=+%283.45%29%5E2+=+11.9
y2+=+%28-1.45%29%5E2+=+2.1025
check:


11.9=+2%28+3.45%29%2B+5 ..........(1)...plug in x1=+3.45andy1+=+11.9
11.9=+6.9+%2B+5
11.9=+11.9+

y=+%28x%29%5E2.........(2)....plug in x2=+-1.45andy2+=+2.1025

2.1025+=+%28-1.45%29%5E2
2.1025+=+2.1025