Solve by Factoring 5x^(5/2)-10x^(3/2)=120x^(1/2)

$5 x^{\frac{5}{2}} - 10 x^{\frac{3}{2}} = 120 x^{\frac{1}{2}}$

Step 1

Subtract $120 x^{\frac{1}{2}}$ from both sides of the equation.

$5 x^{\frac{5}{2}} - 10 x^{\frac{3}{2}} - 120 x^{\frac{1}{2}} = 0$

Step 2

Factor $5 x^{\frac{1}{2}}$ out of $5 x^{\frac{5}{2}} - 10 x^{\frac{3}{2}} - 120 x^{\frac{1}{2}}$ .

Factor $5 x^{\frac{1}{2}}$ out of $5 x^{\frac{5}{2}}$ .

$5 x^{\frac{1}{2}} (x^{\frac{4}{2}}) - 10 x^{\frac{3}{2}} - 120 x^{\frac{1}{2}} = 0$

Factor $5 x^{\frac{1}{2}}$ out of $- 10 x^{\frac{3}{2}}$ .

$5 x^{\frac{1}{2}} (x^{\frac{4}{2}}) + 5 x^{\frac{1}{2}} (- 2 x^{\frac{2}{2}}) - 120 x^{\frac{1}{2}} = 0$

Factor $5 x^{\frac{1}{2}}$ out of $- 120 x^{\frac{1}{2}}$ .

$5 x^{\frac{1}{2}} (x^{\frac{4}{2}}) + 5 x^{\frac{1}{2}} (- 2 x^{\frac{2}{2}}) + 5 x^{\frac{1}{2}} (- 24) = 0$

Factor $5 x^{\frac{1}{2}}$ out of $5 x^{\frac{1}{2}} (x^{\frac{4}{2}}) + 5 x^{\frac{1}{2}} (- 2 x^{\frac{2}{2}})$ .

$5 x^{\frac{1}{2}} (x^{\frac{4}{2}} - 2 x^{\frac{2}{2}}) + 5 x^{\frac{1}{2}} (- 24) = 0$

Factor $5 x^{\frac{1}{2}}$ out of $5 x^{\frac{1}{2}} (x^{\frac{4}{2}} - 2 x^{\frac{2}{2}}) + 5 x^{\frac{1}{2}} (- 24)$ .

$5 x^{\frac{1}{2}} (x^{\frac{4}{2}} - 2 x^{\frac{2}{2}} - 24) = 0$

Step 3

Factor $x^{\frac{4}{2}} - 2 x^{\frac{2}{2}} - 24$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Write the factored form using these integers.

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

Step 4

Divide $2$ by $2$ .

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

Step 5

Divide $2$ by $2$ .

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

Step 6

Remove unnecessary parentheses.

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

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{\frac{1}{2}} = 0$

$x - 6 = 0$

$x + 4 = 0$

Step 8

Set $x^{\frac{1}{2}}$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}}$ equal to $0$ .

$x^{\frac{1}{2}} = 0$

Solve $x^{\frac{1}{2}} = 0$ for $x$ .

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 0^{2}$

Rewrite the expression.

$x^{1} = 0^{2}$

Simplify.

$x = 0^{2}$

Simplify the right side.

Raising $0$ to any positive power yields $0$ .

$x = 0$

Step 9

Set $x - 6$ equal to $0$ and solve for $x$ .

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

Step 10

Set $x + 4$ equal to $0$ and solve for $x$ .

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 11

The final solution is all the values that make $5 x^{\frac{1}{2}} (x - 6) (x + 4) = 0$ true.

$x = 0, 6, - 4$

Step 12

Exclude the solutions that do not make $5 x^{\frac{1}{2}} (x - 6) (x + 4) = 0$ true.

$x = 0, 6$

Solve by Factoring 5x^(5/2)-10x^(3/2)=120x^(1/2)

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