Question
3. \( 4 x^{2}+8 x+3=0 \)
Ask by Sandoval Frank. in South Africa
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions to the equation \( 4x^{2} + 8x + 3 = 0 \) are \( x = -\frac{1}{2} \) and \( x = -\frac{3}{2} \).
Solution
To solve the quadratic equation:
\[
4x^{2} + 8x + 3 = 0
\]
we can use the **quadratic formula**, which is given by:
\[
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
where \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^{2} + bx + c = 0 \).
**Step 1: Identify the coefficients**
\[
a = 4,\quad b = 8,\quad c = 3
\]
**Step 2: Calculate the discriminant (\( \Delta \))**
\[
\Delta = b^{2} - 4ac = (8)^{2} - 4 \times 4 \times 3 = 64 - 48 = 16
\]
**Step 3: Compute the square root of the discriminant**
\[
\sqrt{\Delta} = \sqrt{16} = 4
\]
**Step 4: Apply the quadratic formula**
\[
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-8 \pm 4}{8}
\]
**Step 5: Solve for the two possible values of \( x \)**
1. **First solution (\( + \) sign):**
\[
x = \frac{-8 + 4}{8} = \frac{-4}{8} = -\frac{1}{2}
\]
2. **Second solution (\( - \) sign):**
\[
x = \frac{-8 - 4}{8} = \frac{-12}{8} = -\frac{3}{2}
\]
**Final Answer:**
\[
x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{3}{2}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To solve the equation \( 4x^2 + 8x + 3 = 0 \), you can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = 8 \), and \( c = 3 \). Plugging in these values, we get \( x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4} \). Simplifying, you'll find the roots of this polynomial, giving you the values of \( x \). This particular quadratic might remind you of some classic themes in literature that deal with transformation and change. Just like characters often face obstacles to overcome, solving equations takes you on a journey through numbers and variables to uncover solutions, turning abstract concepts into tangible results!
