+2862 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?
+4609 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.
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
