Solve by Factoring x^(1/2)-8x^(1/4)+15=0

$x^{\frac{1}{2}} - 8 x^{\frac{1}{4}} + 15 = 0$

Step 1

Rewrite $x^{\frac{1}{2}}$ as ${(x^{\frac{1}{4}})}^{2}$ .

${(x^{\frac{1}{4}})}^{2} - 8 x^{\frac{1}{4}} + 15 = 0$

Step 2

Let $u = x^{\frac{1}{4}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{4}}$ .

$u^{2} - 8 u + 15 = 0$

Step 3

Factor $u^{2} - 8 u + 15$ 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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Write the factored form using these integers.

$(u - 5) (u - 3) = 0$

Step 4

Replace all occurrences of $u$ with $x^{\frac{1}{4}}$ .

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

Step 5

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

$x^{\frac{1}{4}} - 5 = 0$

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

Step 6

Set $x^{\frac{1}{4}} - 5$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{4}} - 5$ equal to $0$ .

$x^{\frac{1}{4}} - 5 = 0$

Solve $x^{\frac{1}{4}} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{\frac{1}{4}} = 5$

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = 5^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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

$x^{\frac{1}{4} \cdot 4} = 5^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 5^{4}$

Rewrite the expression.

$x^{1} = 5^{4}$

Simplify.

$x = 5^{4}$

Simplify the right side.

Raise $5$ to the power of $4$ .

$x = 625$

Step 7

Set $x^{\frac{1}{4}} - 3$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{4}} - 3$ equal to $0$ .

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

Solve $x^{\frac{1}{4}} - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$x^{\frac{1}{4}} = 3$

Raise each side of the equation to the power of $4$ to eliminate the fractional exponent on the left side.

${(x^{\frac{1}{4}})}^{4} = 3^{4}$

Simplify the exponent.

Simplify the left side.

Simplify ${(x^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${(x^{\frac{1}{4}})}^{4}$ .

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

$x^{\frac{1}{4} \cdot 4} = 3^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

$x^{\frac{1}{4} \cdot 4} = 3^{4}$

Rewrite the expression.

$x^{1} = 3^{4}$

Simplify.

$x = 3^{4}$

Simplify the right side.

Raise $3$ to the power of $4$ .

$x = 81$

Step 8

The final solution is all the values that make $(x^{\frac{1}{4}} - 5) (x^{\frac{1}{4}} - 3) = 0$ true.

$x = 625, 81$

Solve by Factoring x^(1/2)-8x^(1/4)+15=0

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