Solving the quadratic equation (2x^2 – 3x – 5 = 0) involves finding the values of (x) that satisfy the equation. As with any quadratic equation, which has the general form (ax^2 + bx + c = 0), there are various methods to find the solutions. In this case, we’ll utilize the quadratic formula and explore its application.
Understanding the Quadratic Formula
The quadratic formula is:
[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} ]
This formula calculates the roots of any quadratic equation (ax^2 + bx + c = 0). The term under the square root, (b^2 – 4ac), is known as the discriminant. It helps determine the nature of the roots (whether they are real and distinct, real and equal, or complex).
Also Read: Solve: x*x*x is equal to 2 x
Applying the Quadratic Formula to (2x^2 – 3x – 5 = 0)
- Identify the coefficients: Here, (a = 2), (b = -3), and (c = -5).
- Calculate the discriminant: (b^2 – 4ac = (-3)^2 – 4 \times 2 \times -5).
- Insert the values into the formula: Substitute (a), (b), and (c) into the quadratic formula.
- Compute the solutions: This will result in two values for (x), which are the solutions to the equation.
Also Read: 5x – 12 = 0
Solving the Equation
Now, let’s solve (2x^2 – 3x – 5 = 0):
- Compute the discriminant:
[ \Delta = (-3)^2 – 4 \times 2 \times -5 = 9 + 40 = 49 ] - Apply the quadratic formula:
[ x = \frac{-(-3) \pm \sqrt{49}}{2 \times 2} = \frac{3 \pm 7}{4} ] This gives two solutions:
[ x_1 = \frac{3 + 7}{4} = \frac{10}{4} = 2.5 ]
[ x_2 = \frac{3 – 7}{4} = \frac{-4}{4} = -1 ]
Also Read : 4x ^ 2 – 5x – 12 = 0
Conclusion
The quadratic formula provides a systematic approach to solving quadratic equations, even when they are not easily factorable. In the case of (2x^2 – 3x – 5 = 0), we found two real solutions: (x = 2.5) and (x = -1). Understanding this method is crucial in algebra and forms the basis for many higher-level mathematical concepts. The ability to determine the nature of the roots based on the discriminant is also an important aspect of solving quadratic equations.