Solve for x 3 square root of x+x^2=15+x^2

$3 \sqrt{x} + x^{2} = 15 + x^{2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$3 x^{\frac{1}{2}} + x^{2} = 15 + x^{2}$

Step 2

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

$3 x^{\frac{1}{2}} {(x^{\frac{1}{2}})}^{4} - 15 - 1 {(x^{\frac{1}{2}})}^{4}$

Step 3

Substitute $u$ for $x^{\frac{1}{2}}$ .

$3 (u) {(u)}^{4} - 15 - 1 {(u)}^{4} = 0$

Step 4

Substitute $x$ for $u$ .

$3 (x^{\frac{1}{2}}) {(x^{\frac{1}{2}})}^{4} - 15 - 1 {(x^{\frac{1}{2}})}^{4} = 0$

Step 5

Move all terms containing $x$ to the left side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$3 x^{\frac{1}{2}} + x^{2} - x^{2} = 15$

Combine the opposite terms in $3 x^{\frac{1}{2}} + x^{2} - x^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$3 x^{\frac{1}{2}} + 0 = 15$

Add $3 x^{\frac{1}{2}}$ and $0$ .

$3 x^{\frac{1}{2}} = 15$

Step 6

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

${(3 x^{\frac{1}{2}})}^{2} = 15^{2}$

Step 7

Simplify the exponent.

Simplify the left side.

Simplify ${(3 x^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $3 x^{\frac{1}{2}}$ .

$3^{2} {(x^{\frac{1}{2}})}^{2} = 15^{2}$

Raise $3$ to the power of $2$ .

$9 {(x^{\frac{1}{2}})}^{2} = 15^{2}$

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

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

$9 x^{\frac{1}{2} \cdot 2} = 15^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 x^{\frac{1}{2} \cdot 2} = 15^{2}$

Rewrite the expression.

$9 x^{1} = 15^{2}$

Simplify.

$9 x = 15^{2}$

Simplify the right side.

Raise $15$ to the power of $2$ .

$9 x = 225$

Step 8

Divide each term in $9 x = 225$ by $9$ and simplify.

Divide each term in $9 x = 225$ by $9$ .

$\frac{9 x}{9} = \frac{225}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{225}{9}$

Divide $x$ by $1$ .

$x = \frac{225}{9}$

Simplify the right side.

Divide $225$ by $9$ .

$x = 25$

Solve for x 3 square root of x+x^2=15+x^2

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