Solve x^(1/2)(x^3/2-x^12)

Question

Simplify the expression

\frac{x ^{3} x - 4 x x}{2}

Evaluate

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x \times 2)

Use the commutative property to reorder the terms

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - 2 x)

Subtract the terms

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Simplify

\frac{x ^{3}}{2} - 2 x

Reduce fractions to a common denominator

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

Write all numerators above the common denominator

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

Multiply the terms

\frac{x ^{3} - 4 x}{2}

x^{\frac{1}{2}} \times \frac{x ^{3} - 4 x}{2}

Multiply the terms

\frac{x ^{\frac{1}{2}} ( x ^{3} - 4 x )}{2}

Solution

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Evaluate

x^{\frac{1}{2}} (x^{3} - 4 x)

Use a^{\frac{m}{n}} = n a^{m} to transform the expression

x \times (x^{3} - 4 x)

Multiply each term in the parentheses by x

x \times x^{3} + x \times (- 4 x)

Calculate the product

x^{3} x + x \times (- 4 x)

Calculate the product

x^{3} x - 4 x x

\frac{x ^{3} x - 4 x x}{2}

Show Solution

Find the roots

x_{1} = 0, x_{2} = 2

Evaluate

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x^{1} \times 2)

To find the roots of the expression,set the expression equal to 0

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x^{1} \times 2) = 0

Find the domain

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x^{1} \times 2) = 0, x \geq 0

Calculate

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x^{1} \times 2) = 0

Evaluate the power

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - x \times 2) = 0

Use the commutative property to reorder the terms

x^{\frac{1}{2}} (\frac{x ^{3}}{2} - 2 x) = 0

Subtract the terms

More Steps

Simplify

\frac{x ^{3}}{2} - 2 x

Reduce fractions to a common denominator

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

Write all numerators above the common denominator

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

Multiply the terms

\frac{x ^{3} - 4 x}{2}

x^{\frac{1}{2}} \times \frac{x ^{3} - 4 x}{2} = 0

Multiply the terms

\frac{x ^{\frac{1}{2}} ( x ^{3} - 4 x )}{2} = 0

Simplify

x^{\frac{1}{2}} (x^{3} - 4 x) = 0

Separate the equation into 2 possible cases

x^{\frac{1}{2}} = 0 x^{3} - 4 x = 0

The only way a root could be 0 is when the radicand equals 0

x = 0 x^{3} - 4 x = 0

Solve the equation

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Evaluate

x^{3} - 4 x = 0

Factor the expression

x (x^{2} - 4) = 0

Separate the equation into 2 possible cases

x = 0 x^{2} - 4 = 0

Solve the equation

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Evaluate

x^{2} - 4 = 0

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

x^{2} = 0 + 4

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

x^{2} = 4

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 4

Simplify the expression

x = \pm 2

Separate the equation into 2 possible cases

x = 2 x = - 2

x = 0 x = 2 x = - 2

x = 0 x = 0 x = 2 x = - 2

Find the union

x = 0 x = 2 x = - 2

Check if the solution is in the defined range

x = 0 x = 2 x = - 2, x \geq 0

Find the intersection of the solution and the defined range

x = 0 x = 2

Solution

x_{1} = 0, x_{2} = 2

Show Solution