Solve the equation by making an appropriate substitution.
x minus 17 StartRoot x EndRoot plus 70equals0

We are given the equation:

\[
x - 17\sqrt{x} + 70 = 0
\]

To solve this equation, we can make the substitution \( u = \sqrt{x} \), which means that \( u^2 = x \). Substituting this into the equation gives:

\[
u^2 - 17u + 70 = 0
\]

This is now a quadratic equation in terms of \( u \). We can solve it using the quadratic formula:

\[
u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = -17 \), and \( c = 70 \). Plugging these values into the formula:

\[
u = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(1)(70)}}{2(1)}
\]
\[
u = \frac{17 \pm \sqrt{289 - 280}}{2}
\]
\[
u = \frac{17 \pm \sqrt{9}}{2}
\]
\[
u = \frac{17 \pm 3}{2}
\]

Now, solve for the two possible values of \( u \):

1. \( u = \frac{17 + 3}{2} = 10 \)
2. \( u = \frac{17 - 3}{2} = 7 \)

Since \( u = \sqrt{x} \), we square both values of \( u \) to find \( x \):

1. \( x = 10^2 = 100 \)
2. \( x = 7^2 = 49 \)

Thus, the solutions to the equation are:

\[
x = 100 \quad \text{or} \quad x = 49
\]

Would you like more details on any part of this process?

Here are 5 related questions to explore further:
1. How do you apply the quadratic formula in general?
2. What is the process of making a substitution in solving equations?
3. How can you check the solutions to this equation?
4. What happens when you solve quadratic equations with complex roots?
5. How can you solve a quadratic equation by factoring instead of using the quadratic formula?

**Tip:** Always verify your solutions by plugging them back into the original equation!

Solve the equation by making an appropriate substitution. x minus 17 StartRoot x EndRoot plus 70 equals 0

This guide explains how to solve the equation x - 17√x + 70 = 0 by making a substitution and using the quadratic formula. Learn the step-by-step process, including key algebra concepts such as substitution and solving quadratic equations. Suitable for Grades 8-10.

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