Solve the equation by making an appropriate substitution.
x minus 13 StartRoot x EndRoot plus 30equals0
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Part 1
Make an appropriate substitution and rewrite the equation in quadratic form.

To solve the equation \( x - 13\sqrt{x} + 30 = 0 \), we can make an appropriate substitution to simplify it into a quadratic form.

### Step 1: Substitution
Let:
\[
 u = \sqrt{x} 
\]
This means:
\[
 u^2 = x
\]

### Step 2: Rewrite the equation
Substitute \( u \) into the original equation \( x - 13\sqrt{x} + 30 = 0 \):
\[
 u^2 - 13u + 30 = 0
\]

Now, we have a quadratic equation in terms of \( u \).

### Step 3: Solve the quadratic equation
To solve \( u^2 - 13u + 30 = 0 \), we can use the quadratic formula:
\[
 u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For this equation, \( a = 1 \), \( b = -13 \), and \( c = 30 \).

\[
 u = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(1)(30)}}{2(1)}
\]
\[
 u = \frac{13 \pm \sqrt{169 - 120}}{2}
\]
\[
 u = \frac{13 \pm \sqrt{49}}{2}
\]
\[
 u = \frac{13 \pm 7}{2}
\]

### Step 4: Find the values of \( u \)
\[
 u = \frac{13 + 7}{2} = \frac{20}{2} = 10
\]
\[
 u = \frac{13 - 7}{2} = \frac{6}{2} = 3
\]

### Step 5: Back-substitution
Now, recall

Solve the equation by making an appropriate substitution: x - 13√x + 30 = 0. Part 1: Make an appropriate substitution and rewrite the equation in quadratic form.

Learn how to solve the equation x - 13√x + 30 = 0 by making an appropriate substitution to convert it into a quadratic equation. This problem involves key algebraic concepts, including substitution and the quadratic formula. Suitable for students in Grades 9-11.

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