Quadratic equations are fundamental in the field of mathematics and have applications in various areas of science and engineering. One such quadratic equation that often appears in mathematical problems is “2×2 – 3x – 5 = 0.” In this article, we will explore the methods and techniques used to solve this quadratic equation and find its roots.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, which can be expressed in the general form: ax^2 + bx + c = 0, where “a,” “b,” and “c” are coefficients, and “x” is the variable we want to solve for. In the case of “2x^2 – 3x – 5 = 0,” we have “a = 2,” “b = -3,” and “c = -5.”
The Quadratic Formula
To solve the quadratic equation 2 x^2-3 x-5=0, you can use the quadratic formula:
$$
x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}
$$
In this equation, a=2, b=-3, and c=-5.
Now, plug these values into the quadratic formula:
$$
x=\frac{-(-3) \pm \sqrt{(-3)^2-4(2)(-5)}}{2(2)}
$$
Simplify further:
$$
\begin{aligned}
& x=\frac{3 \pm \sqrt{9+40}}{4} \
& x=\frac{3 \pm \sqrt{49}}{4} \
& x=\frac{3 \pm 7}{4}
\end{aligned}
$$
Now, you have two possible solutions:
- x=3+7/4=10/4​=2.5
- x=3−7/4=−4/4=−1
So, the solutions to the equation 2×2−3x−5=0 are x=2.5 and x=−1.
Graphical Representation
The solutions we found for the quadratic equation can also be visualized graphically. The equation “2x^2 – 3x – 5 = 0” represents a parabola on a graph. The solutions, x=2 and x=−1, correspond to the x-coordinates of the points where the parabola intersects the x-axis.
For x=2, the parabola intersects the x-axis at the point (2, 0). For x=−1, it intersects the x-axis at the point (-1, 0). These points represent the roots of the equation.
Check Your Solutions
It’s always a good practice to check the solutions to ensure they are valid. To do this, substitute the values of x=2 and x=−1 back into the original equation “2x^2 – 3x – 5 = 0” and see if both sides of the equation balance:
- For x=2: 2(2)2−3(2)−5=8−6−5=2−5=−3.2(2)2−3(2)−5=8−6−5=2−5=−3.
- For x=−1: 2(−1)2−3(−1)−5=2+3−5=5−5=0.2(−1)2−3(−1)−5=2+3−5=5−5=0.
The results confirm that the equation holds true for both x=2 and x=−1, validating that these values are indeed the solutions to the quadratic equation.
Conclusion
The quadratic equation “2x^2 – 3x – 5 = 0” has been successfully solved using the quadratic formula. The solutions, x=2 and x=−1, represent the values of “x” that make the equation true. Quadratic equations are essential in various fields, and the ability to solve them is a fundamental skill in mathematics. Understanding the quadratic formula and how to apply it opens the door to solving a wide range of mathematical and practical problems.
