4X^2 – 5X – 12 = 0: Unveiling Its Mathematical Roots!

It seems like there’s a slight discrepancy in the final roots given in the FAQ section compared to the actual calculated roots. Let’s clarify the correct roots based on the calculations:

Solving the Quadratic Equation 4X^2 – 5X – 12 = 0

Step-by-Step Solution:

Step 1: Identify the Constants

• $a=4$
• $b=-5$
• $c=-12$

Step 2: Calculate the Discriminant

• Discriminant Δ=b2−4ac\Delta = b^2 – 4acΔ=b2−4ac
• b2=(−5)2=25b^2 = (-5)^2 = 25b2=(−5)2=25
• $4ac=4\cdot 4\cdot (-12)=-192$
• $\Delta =25-(-192)=25+192=217$

Step 3: Apply the Quadratic Formula

• Quadratic formula: X=−b±Δ2aX = \frac{-b \pm \sqrt{\Delta}}{2a}X=2a−b±Δ​​
• X=−(−5)±2172⋅4X = \frac{-(-5) \pm \sqrt{217}}{2 \cdot 4}X=2⋅4−(−5)±217​​
• X=5±2178X = \frac{5 \pm \sqrt{217}}{8}X=85±217​​

Step 4: Find the Two Roots

• Approximating the roots using a calculator:
  • X1=5+2178≈3.22X_1 = \frac{5 + \sqrt{217}}{8} \approx 3.22X1​=85+217​​≈3.22
  • X2=5−2178≈−1.22X_2 = \frac{5 – \sqrt{217}}{8} \approx -1.22X2​=85−217​​≈−1.22

Summary of the Solution:

Frequently Asked Questions:

1. What Are The Roots Of 4x^2 – 5x – 12 = 0?
   • The roots are X1≈3.22X_1 \approx 3.22X1​≈3.22 and X2≈−1.22X_2 \approx -1.22X2​≈−1.22.
2. How To Solve 4x^2 – 5x – 12 = 0?
   • Use the quadratic formula: X=−b±b2−4ac2aX = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}X=2a−b±b2−4ac​​.
3. What Does 4x^2 – 5x – 12 = 0 Represent?
   • It represents a quadratic equation with two real roots.
4. Why Is 4x^2 – 5x – 12 = 0 Important?
   • It’s important for understanding quadratic equations and their solutions.

Conclusion:

We have successfully solved the quadratic equation 4X2–5X–12=04X^2 – 5X – 12 = 04X2–5X–12=0 using the quadratic formula. The roots, approximately $3.22$ and $-1.22$, help us understand how to find solutions to quadratic equations and their practical applications. Keep practicing to enhance your skills in solving such equations!