Pregunta
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The solutions are \( a = 3 \) and \( a = -17 \).
Solución
To solve the quadratic equation:
\[
a^{2} + 14a - 51 = 0
\]
we can use the **quadratic formula**, which is given by:
\[
a = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
Here, the coefficients from the equation \( a^{2} + 14a - 51 = 0 \) are:
- \( A = 1 \)
- \( B = 14 \)
- \( C = -51 \)
**Step 1: Calculate the Discriminant**
The discriminant (\( D \)) is the part under the square root in the quadratic formula:
\[
D = B^{2} - 4AC = 14^{2} - 4 \times 1 \times (-51) = 196 + 204 = 400
\]
**Step 2: Compute the Square Root of the Discriminant**
\[
\sqrt{D} = \sqrt{400} = 20
\]
**Step 3: Apply the Quadratic Formula**
\[
a = \frac{-14 \pm 20}{2 \times 1} = \frac{-14 \pm 20}{2}
\]
This yields two possible solutions:
1. **First Solution (\(+\) sign):**
\[
a = \frac{-14 + 20}{2} = \frac{6}{2} = 3
\]
2. **Second Solution (\(-\) sign):**
\[
a = \frac{-14 - 20}{2} = \frac{-34}{2} = -17
\]
**Final Answer:**
\[
a = 3 \quad \text{or} \quad a = -17
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Extra Insights
To solve the quadratic equation \( a^{2}+14 a-51=0 \), you can use the quadratic formula: \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 14 \), and \( c = -51 \). Plugging in these values, we find the discriminant \( b^2 - 4ac = 14^2 - 4 \cdot 1 \cdot (-51) = 196 + 204 = 400 \). This means our square root is 20, and the solutions are \( a = \frac{-14 \pm 20}{2} \). So we get two possible solutions: \( a = 3 \) and \( a = -17 \). Voila! To double-check, replace \( a \) back in the original equation. For \( a = 3\): \( 3^2 + 14×3 - 51 = 0 \), and for \( a = -17\): \( (-17)^2 + 14×(-17) - 51 = 0 \). Both solutions satisfy the equation perfectly. 🎉
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