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solvefor x,f(x+h)-f=3x-4x^2

Solution

solve for

x, f (x + h) - f = 3 x - 4 x^{2}

Solution

x = - \frac{- 3 + f + f ^{2} - 16 h f + 10 f + 9}{8}, x = - \frac{- 3 + f - f ^{2} - 16 h f + 10 f + 9}{8}

Solution steps

f (x + h) - f = 3 x - 4 x^{2}

Expand

f (x + h) - f : x f + h f - f

x f + h f - f = 3 x - 4 x^{2}

Switch sides

3 x - 4 x^{2} = x f + h f - f

Move

f

to the left side

3 x - 4 x^{2} + f = x f + h f

Move

h f

to the left side

3 x - 4 x^{2} + f - h f = x f

Move

x f

to the left side

3 x - 4 x^{2} + f - h f - x f = 0

Write in the standard form

a x^{2} + b x + c = 0

- 4 x^{2} + (3 - f) x + f - h f = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( 3 - f ) \pm ( 3 - f ) ^{2} - 4 ( - 4 ) ( f - h f )}{2 ( - 4 )}

Simplify

(3 - f)^{2} - 4 (- 4) (f - h f) : f^{2} - 16 h f + 10 f + 9

x_{1, 2} = \frac{- ( 3 - f ) \pm f ^{2} - 16 h f + 10 f + 9}{2 ( - 4 )}

Separate the solutions

x_{1} = \frac{- ( 3 - f ) + f ^{2} - 16 h f + 10 f + 9}{2 ( - 4 )}, x_{2} = \frac{- ( 3 - f ) - f ^{2} - 16 h f + 10 f + 9}{2 ( - 4 )}

x = \frac{- ( 3 - f ) + f ^{2} - 16 h f + 10 f + 9}{2 ( - 4 )} : - \frac{- 3 + f + f ^{2} - 16 h f + 10 f + 9}{8}

x = \frac{- ( 3 - f ) - f ^{2} - 16 h f + 10 f + 9}{2 ( - 4 )} : - \frac{- 3 + f - f ^{2} - 16 h f + 10 f + 9}{8}

The solutions to the quadratic equation are:

x = - \frac{- 3 + f + f ^{2} - 16 h f + 10 f + 9}{8}, x = - \frac{- 3 + f - f ^{2} - 16 h f + 10 f + 9}{8}

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