Equation 2: Prove that ##x^2+2x\sqrt x+3x+2\sqrt x+1=0##

• MHB
• solakis1
In summary, to solve this equation, you can use the quadratic formula or complete the square method. It can also be simplified by factoring, but the simplified equation may not be quadratic. The equation has two solutions, which can be found using a graphing calculator. The number 2 in the equation is the coefficient of the square root term and affects the shape of the graph and the number of solutions.
solakis1
I checked the following equation with Wolfram\Alpha and the answer was no real solution
How can we prove that?
$x^2+2x\sqrt x+3x+2\sqrt x+1=0$

solakis said:
I checked the following equation with Wolfram\Alpha and the answer was no real solution
How can we prove that?
$x^2+2x\sqrt x+3x+2\sqrt x+1=0$
You pretty much are stuck with two choices: Ferrari's method or graphing. I'd choose graphing!

-Dan

$f(x) = x^2 + 2x\sqrt x + 3x + 2\sqrt x + 1 = (x + \sqrt x + 1)^2$, so if $f(x) = 0$ then $x + \sqrt x + 1 = 0$. That is a quadratic equation for $\sqrt x$, with solutions $\sqrt x = \frac12(-1 \pm i\sqrt3)$. If $\sqrt x$ is non-real then so is $x$. Therefore the equation $f(x) = 0$ has no real solutions.

[sp]Well, if $\sqrt{x}$ is not pure imaginary then x is complex, anyway.[/sp]

Nice catch!

-Dan

ha, ha so easy solution

What is the purpose of "Equation 2"?

"Equation 2" is a mathematical equation that is used to prove that a specific expression, in this case ##x^2+2x\sqrt x+3x+2\sqrt x+1##, is equal to zero. It is often used in algebra and calculus to solve for unknown variables or to demonstrate mathematical concepts.

How do I solve "Equation 2"?

To solve "Equation 2," you can use various methods such as factoring, substitution, or the quadratic formula. The specific method you use will depend on the complexity of the equation and your personal preference. It is important to carefully follow the steps of the chosen method to arrive at the correct solution.

What is the significance of the variables in "Equation 2"?

The variables in "Equation 2" represent unknown quantities that we are trying to solve for. In this particular equation, ##x## is the variable we are trying to find, while ##\sqrt x## represents the square root of x. These variables allow us to manipulate the equation and arrive at a solution.

Can "Equation 2" have multiple solutions?

Yes, "Equation 2" can have multiple solutions. In fact, most equations in mathematics have multiple solutions. However, some equations may only have one solution or no solution at all. It is important to check your solution and make sure it satisfies the original equation.

How can I check if my solution to "Equation 2" is correct?

To check if your solution to "Equation 2" is correct, you can substitute the value of the variable into the original equation and see if it satisfies the equation. If it does, then your solution is correct. You can also use a graphing calculator or online equation solver to verify your solution.

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