What is f(x)=x^2+3f(x)(f(x)) ?

The solution to f(x)=x^2+3f(x)(f(x)) is Ellipse with (h,k)=(0, 1/6),a= 1/(2sqrt(3)),b= 1/6

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f(x)=x^2+3f(x)(f(x))

f (x) = x^{2} + 3 f (x) (f (x))

(h, k) = (0, \frac{1}{6}), a = \frac{1}{2 3}, b = \frac{1}{6}

Solution steps

y = x^{2} + 3 yy

Rewrite

y - x^{2} - 3 yy = 0

in the form of the standard ellipse equation

\frac{( x - 0 ) ^{2}}{( \frac{1}{2 3} ) ^{2}} + \frac{( y - \frac{1}{6} ) ^{2}}{( \frac{1}{6} ) ^{2}} = 1

Therefore ellipse properties are:

(h, k) = (0, \frac{1}{6}), a = \frac{1}{2 3}, b = \frac{1}{6}

a > b

therefore

a

is semi-major axis and

b

is semi-minor axis

Ellipse with center (h, k) = (0, \frac{1}{6}), a = \frac{1}{2 3}, b = \frac{1}{6}

x^{2} + 4 y^{2} - 6 x - 8 y + 9 = 0

16 x^{2} + 36 y^{2} = 49

\frac{( x + 1.23 ) ^{2}}{1.5 9 ^{2}} + \frac{y ^{2}}{1.0 1 ^{2}} = 1

\frac{x ^{2}}{12} + \frac{y ^{2}}{9} = 1

5 x^{2} + 6 y^{2} - 30 x - 24 y + 39 = 0

What is f(x)=x^2+3f(x)(f(x)) ?
The solution to f(x)=x^2+3f(x)(f(x)) is Ellipse with (h,k)=(0, 1/6),a= 1/(2sqrt(3)),b= 1/6