To solve the equation \(4x^2 + 8x + 7 = 4\), follow these steps:

1. **Subtract 4 from both sides** to set the equation to zero:
   \[
   4x^2 + 8x + 7 - 4 = 0
   \]
   \[
   4x^2 + 8x + 3 = 0
   \]

2. **Simplify the equation**:
   \[
   4x^2 + 8x + 3 = 0
   \]

3. **Use the quadratic formula**:
   The quadratic formula is given by:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \(a = 4\), \(b = 8\), and \(c = 3\).

4. **Plug in the values**:
   \[
   x = \frac{-8 \pm \sqrt{8^2 - 4(4)(3)}}{2(4)}
   \]
   \[
   x = \frac{-8 \pm \sqrt{64 - 48}}{8}
   \]
   \[
   x = \frac{-8 \pm \sqrt{16}}{8}
   \]
   \[
   x = \frac{-8 \pm 4}{8}
   \]

5. **Simplify the solutions**:
   - For the plus sign:
     \[
     x = \frac{-8 + 4}{8} = \frac{-4}{8} = -\frac{1}{2}
     \]
   - For the minus sign:
     \[
     x = \frac{-8 - 4}{8} = \frac{-12}{8} = -\frac{3}{2}
     \]

So, the solutions are:
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{3}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How would you solve a similar quadratic equation, but with different coefficients?
2. What does the discriminant (the part under the square root) tell us about the roots?
3. How can you factorize the quadratic equation \(4x^2 + 8x + 3 = 0\) directly?
4. What is the significance of the solutions in a real-world context?
5. How does the quadratic formula derive from completing the square?

**Tip:** Always check if a quadratic equation can be factored easily before applying the quadratic formula; this can save time!

Learn how to solve the quadratic equation 4x² + 8x + 7 = 4 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula and solving equations. Suitable for students in Grades 9-12.

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