+0

+3
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The sum of a number, its square, and its square root is 2457. What is the number?

I solved this problem by pretending it is a quadratic and using the quadratic formula. I also tried other ways like squaring both sides.

I keep getting a really complicated answer, which is definitely wrong.

Can someone teach me how to solve this?

May 17, 2019

#1
0

x^2 + x + sqrt(x) =2457, solve for x

x = 49

May 17, 2019
#2
+2

We have $$x+x^2+\sqrt{x}=2457.$$

The best thing to do is to set boundary parenthesis.

Let's try an integer value of $$x$$, denote 36, which is 6^2. Plugging it in, we get $$36+36^2+6=1338$$, so we have to go a bit higher.

Tring 49 gives us: $$49+49^2+7=2457$$-Yes!

Thus, the answer is $$\boxed{49}.$$

In these type of problems...it's best to set boundaries to minimize the range of the values. You could try moving the $$x$$'s to the other side, but that could go ugly.

May 17, 2019
#3
+2

Solve for x:
-2457 + sqrt(x) + x + x^2 = 0

Isolate the radical to the left hand side.
Subtract x^2 + x - 2457 from both sides:
sqrt(x) = -x^2 - x + 2457

Eliminate the square root on the left hand side.
Raise both sides to the power of two:
x = (-x^2 - x + 2457)^2

Write the quartic polynomial on the right hand side in standard form.
Expand out terms of the right hand side:
x = x^4 + 2 x^3 - 4913 x^2 - 4914 x + 6036849

Move everything to the left hand side.
Subtract x^4 + 2 x^3 - 4913 x^2 - 4914 x + 6036849 from both sides:
-x^4 - 2 x^3 + 4913 x^2 + 4915 x - 6036849 = 0

Factor the left hand side.
The left hand side factors into a product with three terms:
-(x - 49) (x^3 + 51 x^2 - 2414 x - 123201) = 0
Split the above into 2 different equation:
-(x - 49) =0, and
(x^3 + 51 x^2 - 2414 x - 123201) = 0
This cubic equation gives complex roots only.
Therefore: -(x - 49) =0. Multiply both sides by -1
(x - 49) = 0
x = 49

May 17, 2019
edited by Guest  May 17, 2019