Use the quadratic formula to solve. Express your answer in simplest form.
8, c, squared, minus, 14, c, plus, 5, equals, 0
8c 
2
 −14c+5=
0

The equation given is:

\[
8c^2 - 14c + 5 = 0
\]

We will use the **quadratic formula** to solve this. The quadratic formula is:

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

For the equation \(8c^2 - 14c + 5 = 0\), we can identify the coefficients as follows:
- \(a = 8\)
- \(b = -14\)
- \(c = 5\)

Now, let's substitute these values into the quadratic formula:

\[
c = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(8)(5)}}{2(8)}
\]

Simplify the expression step by step:

\[
c = \frac{14 \pm \sqrt{196 - 160}}{16}
\]
\[
c = \frac{14 \pm \sqrt{36}}{16}
\]
\[
c = \frac{14 \pm 6}{16}
\]

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

1. \(c = \frac{14 + 6}{16} = \frac{20}{16} = \frac{5}{4}\)
2. \(c = \frac{14 - 6}{16} = \frac{8}{16} = \frac{1}{2}\)

Thus, the solutions are:

\[
c = \frac{5}{4} \quad \text{or} \quad c = \frac{1}{2}
\]

These are the two values of \(c\) in simplest form.

Would you like further details or clarifications?

Here are five related questions to expand on this:

1. How can the quadratic formula be derived?
2. What are the conditions for a quadratic equation to have real, repeated, or complex roots?
3. Can the quadratic formula be used to solve non-integer coefficients?
4. How do you factor quadratics if the discriminant is a perfect square?
5. What are some real-life applications of solving quadratic equations?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) to determine the nature of the roots before solving.

Use the quadratic formula to solve. Express your answer in simplest form. 8c^2 - 14c + 5 = 0

Solve the quadratic equation 8c^2 - 14c + 5 using the quadratic formula. Follow a detailed step-by-step breakdown, learn about discriminants, and explore both possible solutions. Ideal for students in Grades 9-10.

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