Find All Complex Solutions x-13 square root of x+42=0

$x - 13 \sqrt{x} + 42 = 0$

Step 1

Factor the left side of the equation.

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

$x - 13 x^{\frac{1}{2}} + 42 = 0$

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

${(x^{\frac{1}{2}})}^{2} - 13 x^{\frac{1}{2}} + 42 = 0$

Let $u = x^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{2}}$ .

$u^{2} - 13 u + 42 = 0$

Factor $u^{2} - 13 u + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Write the factored form using these integers.

$(u - 7) (u - 6) = 0$

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

$(x^{\frac{1}{2}} - 7) (x^{\frac{1}{2}} - 6) = 0$

Step 2

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

$x^{\frac{1}{2}} - 7 = 0$

$x^{\frac{1}{2}} - 6 = 0$

Step 3

Set $x^{\frac{1}{2}} - 7$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 7$ equal to $0$ .

$x^{\frac{1}{2}} - 7 = 0$

Solve $x^{\frac{1}{2}} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$x^{\frac{1}{2}} = 7$

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

${(x^{\frac{1}{2}})}^{2} = 7^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 7^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 7^{2}$

Rewrite the expression.

$x^{1} = 7^{2}$

Simplify.

$x = 7^{2}$

Simplify the right side.

Raise $7$ to the power of $2$ .

$x = 49$

Step 4

Set $x^{\frac{1}{2}} - 6$ equal to $0$ and solve for $x$ .

Set $x^{\frac{1}{2}} - 6$ equal to $0$ .

$x^{\frac{1}{2}} - 6 = 0$

Solve $x^{\frac{1}{2}} - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$x^{\frac{1}{2}} = 6$

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

${(x^{\frac{1}{2}})}^{2} = 6^{2}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 6^{2}$

Rewrite the expression.

$x^{1} = 6^{2}$

Simplify.

$x = 6^{2}$

Simplify the right side.

Raise $6$ to the power of $2$ .

$x = 36$

Step 5

The final solution is all the values that make $(x^{\frac{1}{2}} - 7) (x^{\frac{1}{2}} - 6) = 0$ true.

$x = 49, 36$

Find All Complex Solutions x-13 square root of x+42=0

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