Why are the values (-1.618, 0) and (0.618 ,0) solutions for this equation?

[A.T.]

Formula for solving Quadratic Equations does not work for Cubic Equations. So, sad..

You will be really sad when you see the actual cubic formula:
https://en.wikipedia.org/wiki/Cubic_equation#General_cubic_formula

 

[valenumr]

Do you recognize the golden ratio there? The limiting ratio between successive terms of the Fibonacci sequence is the Golden ratio, 1.61803... to 1. The ratio is sometimes referred to as phi ($\phi$)

The polynomial that you are looking at, $x^2 + x - 1$ is the characteristic polynomial for a similar sequence given by $x_{n+2} = x_{n} - x_{n+1}$ which is basically the reverse of the Fibonacci sequence.

Obviously, if you start a Fibonnacci sequence with 0.61803... and 1, the next term will be 1.61803... You may have noticed that $1.61803... = \frac{1}{0.61803...}$. The ratio of consecutive terms in this particular sequence is always a constant equal to $\phi$.

As I recall, if you solve the Fibonacci recurrence you will get some linear combination $k_1 \phi^n + k_2 (-\phi)^{-n}$. For almost any first two terms you use, the limiting ratio of consecutive elements in the resulting sequence will be either $\phi$ or $-\phi$ in both directions.

If you are interested, we can walk through the details of recurrence relations and the methods for solving them.

You could also just use the quadratic formula, $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ on your polynomial: $x^2 + x - 1$ (a = 1, b = 1, c = -1) to get roots of $\frac{\sqrt{5}-1}{2}$ = $\frac{1}{\phi}$ = 0.61803... and $\frac{-\sqrt{5} - 1}{2}$ = $-\phi$ = -1.61803...

That is pretty interesting!

Edit: the Fibonacci part.

 

[pairofstrings]

solve the equation by completing the square

I understand that 'completing the square' is a technique that solves equations like $x^2+x-1=0$. There are many equations with higher exponents, and lower exponents, and to solve them there could be techniques that gives me solutions by satisfying those equations. The 'completing the square' is one such technique for what I mentioned above..

My question is: how to draw such techniques in math, and where to find other such techniques in math. Also if you could, can you please list techniques that forms solution to various other equations here?

 

[jbriggs444]

I understand that 'completing the square' is a technique that solves equations like $x^2+x-1=0$. There are many equations with higher exponents, and lower exponents, and to solve them there could be techniques that gives me solutions by satisfying those equations. The 'completing the square' is one such technique for what I mentioned above..

My question is: how to draw such techniques in math, and where to find other such techniques in math. Also if you could, can you please list techniques that forms solution to various other equations here?

"How do you solve equations in math?" is a pretty broad question. One answer is that you take a math course.

 

[Mark44]

I understand that 'completing the square' is a technique that solves equations like $x^2+x-1=0$. There are many equations with higher exponents, and lower exponents, and to solve them there could be techniques that gives me solutions by satisfying those equations.

The Quadratic Formula could also be used to solve this quadratic equation. The Quadratic Formula can be proved using the method of completing the square.

The advice below from @jbriggs444 is very good, especially given your disappointment that the Quadratic Formula could not be used to solve cubic equations.

One answer is that you take a math course.

 

[SammyS]

Perhaps OP was referring to solving an equation by graphical methods. That is to say, solving $f(x) = g(x)$ by graphing $y=f(x)$ and $y=g(x)$ on the same set of axes, then using the intersections as the solutions.
For this thread, that means graph $y=x^2+x$ and $y=1$, as in the following.

 

[valenumr]

"How do you solve equations in math?" is a pretty broad question. One answer is that you take a math course.

Deserves a mention on the "one liners" thread.

 

[symbolipoint]

One of the members asked a question of @pairofstrings which was not yet answered.
"Do you understand basic algebra"? and "What grade are you in?"

I also ask, exactly which courses have you formally studied in Mathematics up to now, including the course you are currently in?

 

[symbolipoint]

Summary:: I am trying to graph a plot for a simple equation but I am unable to perform the logic.

The equation is x² + x = 1; When I plot the values are (-1.618 , 0) and (0.618 , 0). Why are these numbers emerging from the equation? Can somebody help me with this?

Thanks in advance.

This is one-dimensional only, as written. The level of instruction and course would be Intermediate Algebra.

Review Completing-the-Square from whatever material you have. The value to complete the square is (1/(2*1))^2=1/4.

I will use(mostly) plain text here. You can transcribe it if that helps.

x²+x=1
x²+x+1/4=1+1/4
(x+1/2)²=5/4
x+1/2=+-(1/2)sqrt(5)
x=-1/2+-(1/2)sqrt(5)
x=(-1+-sqrt(5))/2 -----------this is two different values, each of which satisfies the original equation. You can use a decimal approximation for square root of five and find the corresponding x values for your solution.

 

[pairofstrings]

Thank you all for the answers. I figured out how to build equations. Thanks for the support!