Understanding Quadratic Equations:
A quadratic equation typically takes the form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' represents the variable. The equation may have two real roots, one real root, or two complex roots.
Solving 4x^2 – 5x – 12 = 0: To solve the quadratic equation 4x^2 – 5x – 12 = 0, we can use various methods, including factoring, completing the square, or using the quadratic formula. Let's explore the quadratic formula method:
The quadratic formula states that for any quadratic equation ax^2 + bx + c = 0, the solutions for 'x' are given by:
�=−�±�2−4��2�x=2a−b±b2−4ac
Applying this formula to our equation 4x^2 – 5x – 12 = 0, we have: �=4,�=−5,�=−12a=4,b=−5,c=−12
Substituting these values into the quadratic formula: �=−(−5)±(−5)2−4⋅4⋅(−12)2⋅4x=2⋅4−(−5)±(−5)2−4⋅4⋅(−12)
�=5±25+1928x=85±25+192
�=5±2178x=85±217
So, the roots of the equation 4x^2 – 5x – 12 = 0 are given by: �1=5+2178x1=85+217 �2=5−2178x2=85−217
Conclusion: In this article, we discussed the quadratic equation 4x^2 – 5x – 12 = 0 and demonstrated how to solve it using the quadratic formula. Quadratic equations are essential in mathematics and have numerous applications across various disciplines. Mastering the techniques to solve them is crucial for understanding higher-level mathematics and its real-world applications.