Solve each quadratic equation by factoring.

6x2  14x  8 5 0

We are tasked with solving the quadratic equation \(6x^2 - 14x + 8 = 0\) by factoring. Here's how to do it step by step:

1. **Simplify the equation if possible:**
   - We notice that all the coefficients in the equation can be divided by 2 to simplify:
   \[
   6x^2 - 14x + 8 = 0 \quad \text{becomes} \quad 3x^2 - 7x + 4 = 0
   \]
   
2. **Factor the quadratic expression:**
   - We now need to factor \(3x^2 - 7x + 4\).
   - We look for two numbers that multiply to \(3 \times 4 = 12\) and add up to \(-7\). The two numbers are \(-3\) and \(-4\).
   - Rewrite the middle term (\(-7x\)) using these two numbers:
   \[
   3x^2 - 3x - 4x + 4 = 0
   \]
   
3. **Group the terms:**
   - Group the terms in pairs:
   \[
   (3x^2 - 3x) - (4x - 4) = 0
   \]
   
4. **Factor each group:**
   - Factor out the greatest common factor from each group:
   \[
   3x(x - 1) - 4(x - 1) = 0
   \]
   
5. **Factor the entire expression:**
   - Now, factor out the common binomial factor \((x - 1)\):
   \[
   (3x - 4)(x - 1) = 0
   \]
   
6. **Solve for \(x\):**
   - Set each factor equal to zero:
     1. \(3x - 4 = 0 \quad \Rightarrow \quad x = \frac{4}{3}\)
     2. \(x - 1 = 0 \quad \Rightarrow \quad x = 1\)

Thus, the solutions are:
\[
x = 1 \quad \text{or} \quad x = \frac{4}{3}
\]

Would you like more details on any of the steps?

Here are some related questions to explore:
1. What if the quadratic could not be factored? How would you solve it?
2. Can every quadratic equation be solved by factoring?
3. How do you complete the square for a quadratic equation?
4. What is the quadratic formula and when is it used?
5. How would you solve \(3x^2 - 7x + 4 = 0\) graphically?

**Tip:** Always check if you can simplify the equation first by factoring out the greatest common factor (GCF).

Solve each quadratic equation by factoring.
6x^2 - 14x + 8 = 0

Learn how to solve the quadratic equation 6x^2 - 14x + 8 by factoring, including simplification and grouping terms. This problem covers key algebra concepts like the zero product property and factoring methods. Suitable for students in Grades 8-10.

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