Solve for real solutions for the equation $(2x+1)(3x+1)(5x+1)(30x+1)=10$.
Okay, I heard you, why on earth this problem is supposed to be a delicious question that can promote higher mathematics thinking skills?
One of the most common issues math educators are struggling with is the students who underestimate the so-called trivial math problem and they think by the long and typical tedious solving method, the trivial math problem could be safely and successfully solved without a hitch.
Showing posts with label integers. Show all posts
Showing posts with label integers. Show all posts
$3^a+3^b+3^c = 7299$
Find the total number of positive integers ordered pairs of the equation $3^a+3^b+3^c = 7299$.
WLOG, let $c>b>a$.
Rewrite the RHS of the given equation as the product of two factors, we have:
$7299 = 3^a + 3^b + 3^c$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,=3^a\left(1 + \dfrac{3^b}{3^a} +\dfrac{3^c}{3^a}\right)$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,=3^a(1 + 3^{b-a} + 3^{c-a})$
WLOG, let $c>b>a$.
Rewrite the RHS of the given equation as the product of two factors, we have:
$7299 = 3^a + 3^b + 3^c$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,=3^a\left(1 + \dfrac{3^b}{3^a} +\dfrac{3^c}{3^a}\right)$
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,=3^a(1 + 3^{b-a} + 3^{c-a})$
16x²y²-48x²y+24xy²+100x²+16y²-72xy+150x-48y+100=28
Determine the pair(s) of real numbers $(a,\,b)$ that satisfy the equation $16x^2y^2-48x^2y+24xy^2+100x^2+16y^2-72xy+150x-48y+100=28$.
At first glance, this seems like we need to use the modular arithmetic method to solve for $(x,\,y)$ but wait a minute! We need to find not the integers but real for $(x,\,y)$, so nope, modular arithmetic is out of the question...
At first glance, this seems like we need to use the modular arithmetic method to solve for $(x,\,y)$ but wait a minute! We need to find not the integers but real for $(x,\,y)$, so nope, modular arithmetic is out of the question...
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