Solve [ f(x) = \frac{x^2}{x-3} |x-1| ]

Question

Solve the equation

Solve for x

Solve for f

x = 0 x \in (- \infty, 1) \cap x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x \in (- \infty, 1) \cap x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2} x \in [1, + \infty) \cap x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x \in [1, + \infty) \cap x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}

Evaluate

f x = \frac{x ^{2}}{x - 3} ∣ x - 1 ∣

Multiply the terms

f x = \frac{x ^{2} ∣ x - 1 ∣}{x - 3}

Cross multiply

f x (x - 3) = x^{2} ∣ x - 1 ∣

Calculate

More Steps

f x^{2} - 3 f x = x^{2} ∣ x - 1 ∣

Move the expression to the left side

f x^{2} - 3 f x - x^{2} ∣ x - 1 ∣ = 0

Separate the equation into 2 possible cases

f x^{2} - 3 f x - x^{2} (x - 1) = 0, x - 1 \geq 0 f x^{2} - 3 f x - x^{2} (- (x - 1)) = 0, x - 1 < 0

Solve the equation

More Steps

x = 0 x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}, x - 1 \geq 0 f x^{2} - 3 f x - x^{2} (- (x - 1)) = 0, x - 1 < 0

Solve the inequality

More Steps

x = 0 x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}, x \geq 1 f x^{2} - 3 f x - x^{2} (- (x - 1)) = 0, x - 1 < 0

Solve the equation

More Steps

x = 0 x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}, x \geq 1 x = 0 x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2}, x - 1 < 0

Solve the inequality

More Steps

x = 0 x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}, x \geq 1 x = 0 x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2}, x < 1

Find the intersection

x \in [1, + \infty) \cap x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} \cup x \in [1, + \infty) \cap x = \frac{f + 1 - f ^{2} - 10 f + 1}{2} x = 0 x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2}, x < 1

Find the intersection

x \in [1, + \infty) \cap x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} \cup x \in [1, + \infty) \cap x = \frac{f + 1 - f ^{2} - 10 f + 1}{2} x = 0 x \in (- \infty, 1) \cap x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x \in (- \infty, 1) \cap x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2}

Solution

x = 0 x \in (- \infty, 1) \cap x = \frac{- f + 1 + f ^{2} + 10 f + 1}{2} x \in (- \infty, 1) \cap x = \frac{- f + 1 - f ^{2} + 10 f + 1}{2} x \in [1, + \infty) \cap x = \frac{f + 1 + f ^{2} - 10 f + 1}{2} x \in [1, + \infty) \cap x = \frac{f + 1 - f ^{2} - 10 f + 1}{2}

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