Solve \left|3x 9\right| \left|x-3\right|=12

Question

Solve the equation

x_{1} = \frac{9 - 97}{6}, x_{2} = \frac{9 - 65}{6}, x_{3} = \frac{9 + 65}{6}, x_{4} = \frac{9 + 97}{6}

Alternative Form

x_{1} \approx - 0.141476, x_{2} \approx 0.15629, x_{3} \approx 2.84371, x_{4} \approx 3.141476

Evaluate

∣ 3 x \times 9 ∣ ∣ x - 3 ∣ = 12

Simplify

More Steps

Evaluate

∣ 3 x \times 9 ∣ ∣ x - 3 ∣

Multiply the terms

∣ 27 x ∣ ∣ x - 3 ∣

Multiply the terms

∣ 27 x (x - 3) ∣

∣ 27 x (x - 3) ∣ = 12

Separate the equation into 2 possible cases

27 x (x - 3) = 12 27 x (x - 3) = - 12

Solve the equation for x

More Steps

Evaluate

27 x (x - 3) = 12

Expand the expression

More Steps

Evaluate

27 x (x - 3)

Apply the distributive property

27 x \times x - 27 x \times 3

Multiply the terms

27 x^{2} - 27 x \times 3

Multiply the numbers

27 x^{2} - 81 x

27 x^{2} - 81 x = 12

Move the expression to the left side

27 x^{2} - 81 x - 12 = 0

Substitute a= 27,b= - 81 and c= - 12 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{81 \pm ( - 81 ) ^{2} - 4 \times 27 ( - 12 )}{2 \times 27}

Simplify the expression

x = \frac{81 \pm ( - 81 ) ^{2} - 4 \times 27 ( - 12 )}{54}

Simplify the expression

More Steps

Evaluate

(- 81)^{2} - 4 \times 27 (- 12)

Multiply

(- 81)^{2} - (- 1296)

Rewrite the expression

8 1^{2} - (- 1296)

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

8 1^{2} + 1296

Evaluate the power

6561 + 1296

Add the numbers

7857

x = \frac{81 \pm 7857}{54}

Simplify the radical expression

More Steps

Evaluate

7857

Write the expression as a product where the root of one of the factors can be evaluated

81 \times 97

Write the number in exponential form with the base of 9

9^{2} \times 97

The root of a product is equal to the product of the roots of each factor

9^{2} \times 97

Reduce the index of the radical and exponent with 2

997

x = \frac{81 \pm 9 97}{54}

Separate the equation into 2 possible cases

x = \frac{81 + 9 97}{54} x = \frac{81 - 9 97}{54}

Simplify the expression

x = \frac{9 + 97}{6} x = \frac{81 - 9 97}{54}

Simplify the expression

x = \frac{9 + 97}{6} x = \frac{9 - 97}{6}

x = \frac{9 + 97}{6} x = \frac{9 - 97}{6} 27 x (x - 3) = - 12

Solve the equation for x

More Steps

Evaluate

27 x (x - 3) = - 12

Expand the expression

More Steps

Evaluate

27 x (x - 3)

Apply the distributive property

27 x \times x - 27 x \times 3

Multiply the terms

27 x^{2} - 27 x \times 3

Multiply the numbers

27 x^{2} - 81 x

27 x^{2} - 81 x = - 12

Move the expression to the left side

27 x^{2} - 81 x - (- 12) = 0

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

27 x^{2} - 81 x + 12 = 0

Substitute a= 27,b= - 81 and c= 12 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{81 \pm ( - 81 ) ^{2} - 4 \times 27 \times 12}{2 \times 27}

Simplify the expression

x = \frac{81 \pm ( - 81 ) ^{2} - 4 \times 27 \times 12}{54}

Simplify the expression

More Steps

Evaluate

(- 81)^{2} - 4 \times 27 \times 12

Multiply the terms

(- 81)^{2} - 1296

Rewrite the expression

8 1^{2} - 1296

Evaluate the power

6561 - 1296

Subtract the numbers

5265

x = \frac{81 \pm 5265}{54}

Simplify the radical expression

More Steps

Evaluate

5265

Write the expression as a product where the root of one of the factors can be evaluated

81 \times 65

Write the number in exponential form with the base of 9

9^{2} \times 65

The root of a product is equal to the product of the roots of each factor

9^{2} \times 65

Reduce the index of the radical and exponent with 2

965

x = \frac{81 \pm 9 65}{54}

Separate the equation into 2 possible cases

x = \frac{81 + 9 65}{54} x = \frac{81 - 9 65}{54}

Simplify the expression

x = \frac{9 + 65}{6} x = \frac{81 - 9 65}{54}

Simplify the expression

x = \frac{9 + 65}{6} x = \frac{9 - 65}{6}

x = \frac{9 + 97}{6} x = \frac{9 - 97}{6} x = \frac{9 + 65}{6} x = \frac{9 - 65}{6}

Solution

x_{1} = \frac{9 - 97}{6}, x_{2} = \frac{9 - 65}{6}, x_{3} = \frac{9 + 65}{6}, x_{4} = \frac{9 + 97}{6}

Alternative Form

x_{1} \approx - 0.141476, x_{2} \approx 0.15629, x_{3} \approx 2.84371, x_{4} \approx 3.141476

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