Solve || x-3 |-5| = |x^2|

Question

Solve the equation

x_{1} = - 1, x_{2} = 2

Evaluate

∣ ∣ x - 3 ∣ - 5 ∣ = x^{2}

When the expression in absolute value bars is not negative, remove the bars

∣ ∣ x - 3 ∣ - 5 ∣ = x^{2}

Rewrite the expression

∣ ∣ x - 3 ∣ - 5 ∣ - x^{2} = 0

Separate the equation into 2 possible cases

∣ x - 3 ∣ - 5 - x^{2} = 0, ∣ x - 3 ∣ - 5 \geq 0 - (∣ x - 3 ∣ - 5) - x^{2} = 0, ∣ x - 3 ∣ - 5 < 0

Solve the equation

More Steps

Evaluate

∣ x - 3 ∣ - 5 - x^{2} = 0

Separate the equation into 2 possible cases

x - 3 - 5 - x^{2} = 0, x - 3 \geq 0 - (x - 3) - 5 - x^{2} = 0, x - 3 < 0

Solve the equation

More Steps

Evaluate

x - 3 - 5 - x^{2} = 0

Calculate

x - 8 - x^{2} = 0

Rewrite in standard form

- x^{2} + x - 8 = 0

Multiply both sides

x^{2} - x + 8 = 0

Substitute a= 1,b= - 1 and c= 8 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{1 \pm ( - 1 ) ^{2} - 4 \times 8}{2}

Simplify the expression

x = \frac{1 \pm - 31}{2}

The expression is undefined in the set of real numbers

x \in / R

x \in / R, x - 3 \geq 0 - (x - 3) - 5 - x^{2} = 0, x - 3 < 0

Solve the inequality

More Steps

Evaluate

x - 3 \geq 0

Move the constant to the right side

x \geq 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x \geq 3

x \in / R, x \geq 3 - (x - 3) - 5 - x^{2} = 0, x - 3 < 0

Solve the equation

More Steps

Evaluate

- (x - 3) - 5 - x^{2} = 0

Calculate

- x + 3 - 5 - x^{2} = 0

Calculate

- x - 2 - x^{2} = 0

Rewrite in standard form

- x^{2} - x - 2 = 0

Multiply both sides

x^{2} + x + 2 = 0

Substitute a= 1,b= 1 and c= 2 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 1 \pm 1 ^{2} - 4 \times 2}{2}

Simplify the expression

x = \frac{- 1 \pm - 7}{2}

The expression is undefined in the set of real numbers

x \in / R

x \in / R, x \geq 3 x \in / R, x - 3 < 0

Solve the inequality

More Steps

Evaluate

x - 3 < 0

Move the constant to the right side

x < 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x < 3

x \in / R, x \geq 3 x \in / R, x < 3

Find the intersection

x \in / R x \in / R, x < 3

Find the intersection

x \in / R x \in / R

Find the union

x \in / R

x \in / R, ∣ x - 3 ∣ - 5 \geq 0 - (∣ x - 3 ∣ - 5) - x^{2} = 0, ∣ x - 3 ∣ - 5 < 0

Solve the inequality

More Steps

Evaluate

∣ x - 3 ∣ - 5 \geq 0

Rearrange the terms

∣ x - 3 ∣ \geq 5

Separate the inequality into 2 possible cases

x - 3 \geq 5 x - 3 \leq - 5

Solve the inequality for x

More Steps

Evaluate

x - 3 \geq 5

Move the constant to the right side

x \geq 5 + 3

Add the numbers

x \geq 8

x \geq 8 x - 3 \leq - 5

Solve the inequality for x

More Steps

Evaluate

x - 3 \leq - 5

Move the constant to the right side

x \leq - 5 + 3

Add the numbers

x \leq - 2

x \geq 8 x \leq - 2

Find the union

x \in (- \infty, - 2] \cup [8, + \infty)

x \in / R, x \in (- \infty, - 2] \cup [8, + \infty) - (∣ x - 3 ∣ - 5) - x^{2} = 0, ∣ x - 3 ∣ - 5 < 0

Solve the equation

More Steps

Evaluate

- (∣ x - 3 ∣ - 5) - x^{2} = 0

Calculate

- ∣ x - 3 ∣ + 5 - x^{2} = 0

Separate the equation into 2 possible cases

- (x - 3) + 5 - x^{2} = 0, x - 3 \geq 0 - (- (x - 3)) + 5 - x^{2} = 0, x - 3 < 0

Solve the equation

More Steps

Evaluate

- (x - 3) + 5 - x^{2} = 0

Calculate

- x + 3 + 5 - x^{2} = 0

Calculate

- x + 8 - x^{2} = 0

Rewrite in standard form

- x^{2} - x + 8 = 0

Multiply both sides

x^{2} + x - 8 = 0

Substitute a= 1,b= 1 and c= - 8 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 1 \pm 1 ^{2} - 4 ( - 8 )}{2}

Simplify the expression

x = \frac{- 1 \pm 33}{2}

Separate the equation into 2 possible cases

x = \frac{- 1 + 33}{2} x = \frac{- 1 - 33}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x = \frac{- 1 + 33}{2} x = - \frac{1 + 33}{2}

x = \frac{- 1 + 33}{2} x = - \frac{1 + 33}{2}, x - 3 \geq 0 - (- (x - 3)) + 5 - x^{2} = 0, x - 3 < 0

Solve the inequality

More Steps

Evaluate

x - 3 \geq 0

Move the constant to the right side

x \geq 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x \geq 3

x = \frac{- 1 + 33}{2} x = - \frac{1 + 33}{2}, x \geq 3 - (- (x - 3)) + 5 - x^{2} = 0, x - 3 < 0

Solve the equation

More Steps

Evaluate

- (- (x - 3)) + 5 - x^{2} = 0

Calculate

x - 3 + 5 - x^{2} = 0

Calculate

x + 2 - x^{2} = 0

Factor the expression

(2 - x) (1 + x) = 0

When the product of factors equals 0,at least one factor is 0

2 - x = 0 1 + x = 0

Solve the equation for x

x = 2 1 + x = 0

Solve the equation for x

x = 2 x = - 1

x = \frac{- 1 + 33}{2} x = - \frac{1 + 33}{2}, x \geq 3 x = 2 x = - 1, x - 3 < 0

Solve the inequality

More Steps

Evaluate

x - 3 < 0

Move the constant to the right side

x < 0 + 3

Removing 0 doesn't change the value,so remove it from the expression

x < 3

x = \frac{- 1 + 33}{2} x = - \frac{1 + 33}{2}, x \geq 3 x = 2 x = - 1, x < 3

Find the intersection

x \in \emptyset x = 2 x = - 1, x < 3

Find the intersection

x \in \emptyset x = 2 x = - 1

Find the union

x = 2 x = - 1

x \in / R, x \in (- \infty, - 2] \cup [8, + \infty) x = 2 x = - 1, ∣ x - 3 ∣ - 5 < 0

Solve the inequality

More Steps

Evaluate

∣ x - 3 ∣ - 5 < 0

Rearrange the terms

∣ x - 3 ∣ < 5

Separate the inequality into 2 possible cases

{x - 3 < 5 x - 3 > - 5

Solve the inequality for x

More Steps

Evaluate

x - 3 < 5

Move the constant to the right side

x < 5 + 3

Add the numbers

x < 8

{x < 8 x - 3 > - 5

Solve the inequality for x

More Steps

Evaluate

x - 3 > - 5

Move the constant to the right side

x > - 5 + 3

Add the numbers

x > - 2

{x < 8 x > - 2

Find the intersection

- 2 < x < 8

x \in / R, x \in (- \infty, - 2] \cup [8, + \infty) x = 2 x = - 1, - 2 < x < 8

Find the intersection

x \in / R x = 2 x = - 1, - 2 < x < 8

Find the intersection

x \in / R x = 2 x = - 1

Find the union

x = 2 x = - 1

Solution

x_{1} = - 1, x_{2} = 2

Show Solution

Solve the equation

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