Use the quadratic formula to solve the following equation. If the solutions involve square​ roots, give both the exact and approximate solutions.

11 m squared minus 2 m equals 10




We are given the quadratic equation:

\[
11m^2 - 2m = 10
\]

First, rewrite the equation in standard form \( ax^2 + bx + c = 0 \) by moving all terms to one side:

\[
11m^2 - 2m - 10 = 0
\]

Now, we can apply the quadratic formula, which is given by:

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

In this case, \( a = 11 \), \( b = -2 \), and \( c = -10 \). Let's substitute these values into the formula:

\[
m = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(11)(-10)}}{2(11)}
\]

Simplifying:

\[
m = \frac{2 \pm \sqrt{4 + 440}}{22}
\]

\[
m = \frac{2 \pm \sqrt{444}}{22}
\]

Now, express the exact solutions:

\[
m = \frac{2 \pm \sqrt{444}}{22}
\]

To approximate the solutions, find the square root of 444:

\[
\sqrt{444} \approx 21.07
\]

Substitute this approximation into the formula:

\[
m = \frac{2 \pm 21.07}{22}
\]

Now, calculate the two possible solutions:

1. \( m = \frac{2 + 21.07}{22} \approx \frac{23.07}{22} \approx 1.05 \)
2. \( m = \frac{2 - 21.07}{22} \approx \frac{-19.07}{22} \approx -0.87 \)

### Final solutions:

- Exact solutions: 
  \[
  m = \frac{2 + \sqrt{444}}{22}, \quad m = \frac{2 - \sqrt{444}}{22}
  \]

- Approximate solutions:
  \[
  m \approx 1.05, \quad m \approx -0.87
  \]

Would you like further details on this solution?

### Related Questions:
1. How can the quadratic formula be derived?
2. What are the conditions under which a quadratic equation has real solutions?
3. How does the discriminant affect the nature of solutions?
4. What are other methods to solve quadratic equations besides the quadratic formula?
5. Can the quadratic formula be used for non-real solutions?

### Tip:
Always check the discriminant \((b^2 - 4ac)\) to determine if the solutions are real or complex before solving the quadratic equation.

Use the quadratic formula to solve the following equation. If the solutions involve square roots, give both the exact and approximate solutions.

11m^2 - 2m = 10

Learn how to solve the quadratic equation 11m^2 - 2m = 10 using the quadratic formula. This detailed solution explains the process of finding both exact and approximate roots, covering important algebraic concepts and the quadratic formula theorem. Suitable for students in Grades 9-11.

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