Solve |2-x| |x-7|=36

Question

Solve the equation

x_{1} = - 2, x_{2} = 11

Evaluate

∣ 2 - x ∣ ∣ x - 7 ∣ = 36

Multiply the terms

∣ (2 - x) (x - 7) ∣ = 36

Separate the equation into 2 possible cases

(2 - x) (x - 7) = 36 (2 - x) (x - 7) = - 36

Solve the equation for x

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Evaluate

(2 - x) (x - 7) = 36

Expand the expression

More Steps

Evaluate

(2 - x) (x - 7)

Apply the distributive property

2 x - 2 \times 7 - x \times x - (- x \times 7)

Multiply the numbers

2 x - 14 - x \times x - (- x \times 7)

Multiply the terms

2 x - 14 - x^{2} - (- x \times 7)

Use the commutative property to reorder the terms

2 x - 14 - x^{2} - (- 7 x)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

2 x - 14 - x^{2} + 7 x

Add the terms

9 x - 14 - x^{2}

9 x - 14 - x^{2} = 36

Move the expression to the left side

9 x - 14 - x^{2} - 36 = 0

Subtract the numbers

9 x - 50 - x^{2} = 0

Rewrite in standard form

- x^{2} + 9 x - 50 = 0

Multiply both sides

x^{2} - 9 x + 50 = 0

Substitute a= 1,b= - 9 and c= 50 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{9 \pm ( - 9 ) ^{2} - 4 \times 50}{2}

Simplify the expression

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Evaluate

(- 9)^{2} - 4 \times 50

Multiply the numbers

(- 9)^{2} - 200

Rewrite the expression

9^{2} - 200

Evaluate the power

81 - 200

Subtract the numbers

- 119

x = \frac{9 \pm - 119}{2}

The expression is undefined in the set of real numbers

x \in / R

x \in / R (2 - x) (x - 7) = - 36

Solve the equation for x

More Steps

Evaluate

(2 - x) (x - 7) = - 36

Expand the expression

More Steps

Evaluate

(2 - x) (x - 7)

Apply the distributive property

2 x - 2 \times 7 - x \times x - (- x \times 7)

Multiply the numbers

2 x - 14 - x \times x - (- x \times 7)

Multiply the terms

2 x - 14 - x^{2} - (- x \times 7)

Use the commutative property to reorder the terms

2 x - 14 - x^{2} - (- 7 x)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

2 x - 14 - x^{2} + 7 x

Add the terms

9 x - 14 - x^{2}

9 x - 14 - x^{2} = - 36

Move the expression to the left side

9 x - 14 - x^{2} - (- 36) = 0

Subtract the numbers

More Steps

Evaluate

- 14 - (- 36)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

- 14 + 36

Add the numbers

22

9 x + 22 - x^{2} = 0

Factor the expression

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Evaluate

9 x + 22 - x^{2}

Reorder the terms

22 + 9 x - x^{2}

Rewrite the expression

22 + (11 - 2) x - x^{2}

Calculate

22 + 11 x - 2 x - x^{2}

Rewrite the expression

11 \times 2 + 11 x - x \times 2 - x \times x

Factor out 11 from the expression

11 (2 + x) - x \times 2 - x \times x

Factor out - x from the expression

11 (2 + x) - x (2 + x)

Factor out 2 + x from the expression

(11 - x) (2 + x)

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

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

11 - x = 0 2 + x = 0

Solve the equation for x

More Steps

Evaluate

11 - x = 0

Move the constant to the right-hand side and change its sign

- x = 0 - 11

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

- x = - 11

Change the signs on both sides of the equation

x = 11

x = 11 2 + x = 0

Solve the equation for x

More Steps

Evaluate

2 + x = 0

Move the constant to the right-hand side and change its sign

x = 0 - 2

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

x = - 2

x = 11 x = - 2

x \in / R x = 11 x = - 2

Find the union

x = 11 x = - 2

Solution

x_{1} = - 2, x_{2} = 11

Show Solution

Solve the equation

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