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The quadratic equation we are analyzing is $\mathrm{}$2×2-3x- 5 = 0. It’s a fundamental type of polynomial equation, specifically of the second order, and is of great significance in various mathematical and practical applications. The general form of a quadratic equation is **$$**, where $$, $$, and $$ are coefficients and $a\mathrm{\ne}0$. In our specific equation, these coefficients are $a=2$, $$, and $c=−5$.

## Characteristics of the Equation 2×2-3x- 5 = 0

**Degree:**The degree of this equation is 2, which is the highest power of x present in the equation. This is characteristic of all quadratic equations.**Graphical Representation:**Graphically, quadratic equations are represented by a parabola. In this case, since the coefficient of x${}^{2}$ is positive (2), the parabola opens upwards.**Symmetry:**The axis of symmetry of a parabola in a quadratic equation is given by**x**$$**.**This line divides the parabola into two symmetrical halves.**Vertex:**The vertex of a parabola is a significant point, providing the maximum or minimum value of the quadratic function, depending on the direction in which the parabola opens.

## Methods of Solving the Equation

### 1. Factoring

Factoring involves breaking down the quadratic into a product of binomials. This method, however, is not always straightforward and may not be feasible for all quadratics, especially when the roots are not rational.

### 2. Completing the Square

Completing the square is another method where the equation is transformed into a perfect square trinomial. This method can be more complex and time-consuming, especially for beginners.

### 3. Quadratic Formula

The quadratic formula is the most reliable method for solving any quadratic equation. It states that for any quadratic equation

**$ax_{2}+bx+c=0$**, the values of x can be found using: **x$=\frac{-b\pm \sqrt{{b}^{2}-4\mathrm{ac}}}{2a}$ .** This formula is derived from completing the square of the general quadratic equation and is applicable universally.

### 4. Solving 2×2-3x- 5 = 0

Applying the quadratic formula, we proceed as follows:

#### Discriminant Analysis

The discriminant ($\mathrm{}$) of the equation 2×2-3x- 5 = 0 is calculated as $Δ=b_{2}−4ac$. Substituting the values $a=2$, $$, and $$, we find that $\mathrm{\Delta}=49$. Since $\mathrm{\Delta}>0$, this indicates that there are two real and distinct solutions.

#### Finding Roots

Using the quadratic formula, the roots are given by: x$=\frac{-(-3)\pm \sqrt{49}}{2\times 2}$ . Simplifying this, we find two solutions for x.

- ${x}_{1}=-1$
- ${x}_{2}=\frac{5}{2}$

These values are the roots of the quadratic equation, meaning they are the points where the parabola is represented by$\mathrm{}$ 2×2-3x- 5 = 0 intersects the x-axis.

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### Conclusion

Upon solving, we find that the quadratic equation $\mathrm{}$2×2-3x- 5 = 0 has two real and distinct solutions: -1 and $\frac{5}{2}$. These solutions are the x-coordinates where the parabola intersects the x-axis. The significance of these solutions extends beyond mathematics, as quadratic equations are frequently used to model various physical phenomena, in fields ranging from physics and engineering to finance and economics. Understanding the solutions of such equations is therefore crucial for both theoretical and practical applications.