The solutions to the equation 2×2+x-1=2 are x=-3/2 or x=_
The Correct Answer and Explanation is :
To solve the equation (2x^2 + x – 1 = 2), we first rearrange it to standard quadratic form:
[
2x^2 + x – 1 – 2 = 0
]
This simplifies to:
[
2x^2 + x – 3 = 0
]
Next, we will use the quadratic formula to find the solutions. The quadratic formula is given by:
[
x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}
]
Here, (a = 2), (b = 1), and (c = -3). We start by calculating the discriminant (b^2 – 4ac):
[
b^2 – 4ac = 1^2 – 4 \cdot 2 \cdot (-3) = 1 + 24 = 25
]
Since the discriminant is positive, we have two real and distinct solutions. Now we plug the values into the quadratic formula:
[
x = \frac{-1 \pm \sqrt{25}}{2 \cdot 2}
]
This simplifies to:
[
x = \frac{-1 \pm 5}{4}
]
Now we calculate the two possible values for (x):
- For the positive case:
[
x = \frac{-1 + 5}{4} = \frac{4}{4} = 1
] - For the negative case:
[
x = \frac{-1 – 5}{4} = \frac{-6}{4} = -\frac{3}{2}
]
Thus, the solutions to the equation (2x^2 + x – 1 = 2) are:
[
x = 1 \quad \text{and} \quad x = -\frac{3}{2}
]
Explanation
The quadratic formula is a powerful tool for solving any quadratic equation, which is an equation of the form (ax^2 + bx + c = 0). In this case, we first rearranged the original equation into standard form. By calculating the discriminant, we determined the nature of the roots, which in this scenario indicated that there would be two real solutions.
When applying the formula, it’s important to carefully manage the arithmetic operations, especially when simplifying expressions. Each solution derived from the quadratic formula represents a point where the corresponding quadratic function intersects the x-axis, illustrating the fundamental relationship between algebraic equations and their graphical representations. Thus, we find that the equation yields two specific solutions: (x = 1) and (x = -\frac{3}{2}).