Solve 15x^( 5/2 )-2x^( 3/2 )-24x^( 1/2 )

Question

Simplify the expression

15 x^{2} x - 2 x x - 24 x

Evaluate

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}}

Solution

15 x^{2} x - 2 x x - 24 x

Show Solution

x \times (3 x - 4) (5 x + 6)

Evaluate

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}}

Calculate

15 x^{2} x - 2 x x - 24 x

Rewrite the expression

15 x \times x^{2} - 2 x x - 24 x

Evaluate

15 x \times x^{2} - 2 x \times x - 24 x

Rewrite the expression

x \times 15 x^{2} - x \times 2 x - x \times 24

Factor out x from the expression

x \times (15 x^{2} - 2 x - 24)

Solution

More Steps

Evaluate

15 x^{2} - 2 x - 24

Rewrite the expression

15 x^{2} + (18 - 20) x - 24

Calculate

15 x^{2} + 18 x - 20 x - 24

Rewrite the expression

3 x \times 5 x + 3 x \times 6 - 4 \times 5 x - 4 \times 6

Factor out 3 x from the expression

3 x (5 x + 6) - 4 \times 5 x - 4 \times 6

Factor out - 4 from the expression

3 x (5 x + 6) - 4 (5 x + 6)

Factor out 5 x + 6 from the expression

(3 x - 4) (5 x + 6)

x \times (3 x - 4) (5 x + 6)

Show Solution

x_{1} = 0, x_{2} = \frac{4}{3}

Alternative Form

x_{1} = 0, x_{2} = 1. \dot{3}

Evaluate

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}}

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

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}} = 0

Find the domain

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}} = 0, x \geq 0

Calculate

15 x^{\frac{5}{2}} - 2 x^{\frac{3}{2}} - 24 x^{\frac{1}{2}} = 0

Factor the expression

x^{\frac{1}{2}} (3 x - 4) (5 x + 6) = 0

Separate the equation into 3 possible cases

x^{\frac{1}{2}} = 0 3 x - 4 = 0 5 x + 6 = 0

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

x = 0 3 x - 4 = 0 5 x + 6 = 0

Solve the equation

More Steps

Evaluate

3 x - 4 = 0

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

3 x = 0 + 4

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

3 x = 4

Divide both sides

\frac{3 x}{3} = \frac{4}{3}

Divide the numbers

x = \frac{4}{3}

x = 0 x = \frac{4}{3} 5 x + 6 = 0

Solve the equation

More Steps

Evaluate

5 x + 6 = 0

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

5 x = 0 - 6

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

5 x = - 6

Divide both sides

\frac{5 x}{5} = \frac{- 6}{5}

Divide the numbers

x = \frac{- 6}{5}

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

x = - \frac{6}{5}

x = 0 x = \frac{4}{3} x = - \frac{6}{5}

Check if the solution is in the defined range

x = 0 x = \frac{4}{3} x = - \frac{6}{5}, x \geq 0

Find the intersection of the solution and the defined range

x = 0 x = \frac{4}{3}

Solution

x_{1} = 0, x_{2} = \frac{4}{3}

Alternative Form

x_{1} = 0, x_{2} = 1. \dot{3}

Show Solution