To solve the quadratic equation $5x^2 + 14x - 3 = 0$ using the factoring method, follow these steps:

Step 1: Multiply the coefficient of $x^2$ and the constant term.

• Multiply the coefficient of $x^2$ (which is 5) by the constant term (which is -3):

  $5 \times (-3) = -15$

Step 2: Find two numbers that multiply to -15 and add up to the coefficient of $x$ (which is 14).

• We need two numbers that multiply to -15 and add to 14. Let's denote these numbers as $m$ and $n$.

• After checking the pairs, we see that 15 and -1 work because:

  $15 \times (-1) = -15$

  $15 + (-1) = 14$

Step 3: Rewrite the middle term using the numbers found.

• Rewrite the equation by splitting the middle term (14x) into two terms using 15 and -1:

  $5x^2 + 15x - x - 3 = 0$

Step 4: Factor by grouping.

• Group the terms into two pairs:

  $(5x^2 + 15x) - (x + 3) = 0$

• Factor out the common factors from each group:

  $5x(x + 3) - 1(x + 3) = 0$

Step 5: Factor out the common binomial.

• Notice that $(x + 3)$ is common in both terms:

  $(5x - 1)(x + 3) = 0$

Step 6: Solve for $x$.

• Set each factor equal to zero and solve for $x$:

  1. $5x - 1 = 0$ $5x = 1 \implies x = \frac{1}{5}$

  2. $x + 3 = 0$ $x = -3$

Final Answer:

The solutions to the equation $5x^2 + 14x - 3 = 0$ are:

$x = \frac{1}{5} \quad \text{or} \quad x = -3$

Would you like any further explanation or details on this solution?

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Follow-Up Questions:

1. How can the factoring method be applied to other quadratic equations?
2. What is the significance of the Discriminant in quadratic equations?
3. Can every quadratic equation be solved by factoring?
4. How does the quadratic formula compare to factoring?
5. What is the relationship between the roots and the factors of a quadratic equation?
6. How would you solve $2x^2 + 7x - 3 = 0$ by factoring?
7. Why might factoring fail, and what should you do in such cases?
8. What are the next steps after finding the solutions to a quadratic equation?

Tip:

When factoring quadratic equations, always check the product and sum of the factors to ensure they match the original quadratic equation.