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Quadratic equations are mathematical expressions that involve variables raised to the power of two. Typically, they take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not zero. This form is known as the standard form of a quadratic equation.

Quadratic equations can be solved using different methods, including:

• Factoring
• Completing the square
• Using the quadratic formula

Understanding the characteristics of quadratic equations is essential for solving problems involving curves or parabolas. The equation in our example, \( 2x^2 - 8x + 13 = 0 \), is already arranged in standard form, which makes it convenient for identifying the coefficients needed for the substitution method and quadratic formula.

The substitution method is a technique used to solve systems of equations where one equation is used to express one variable in terms of the other. This substitution is then used in the second equation to find the solution. This method is particularly useful when dealing with simultaneous equations, where at least one equation is linear.

In the given exercise, we solve the linear equation \( x + y = 4 \) for \( y \), finding that \( y = 4 - x \). This expression gets substituted into the quadratic equation \( x^2 + y^2 = 3 \).

This substitution simplifies the first equation to involve only one variable, making it easier to solve. Although it didn't lead to a real solution in this scenario, substitution remains a vital tool for solving systems of equations.

The quadratic formula is a versatile tool for finding solutions to quadratic equations when other methods, like factoring, are not feasible. The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) correspond to the coefficients of the quadratic equation in standard form \( ax^2 + bx + c = 0 \).

In the solution of our system, the equation \( 2x^2 - 8x + 13 = 0 \) is solved for \( x \) using the quadratic formula. Plugging the relevant values into the formula reveals a negative number under the square root. This indicates an important concept related to solutions: when the discriminant, \( b^2 - 4ac \), is negative, the quadratic equation has no real roots.

The scenario with a negative discriminant highlights a significant aspect of quadratic equations—when \( b^2 - 4ac < 0 \), the solutions are not real numbers but rather involve imaginary numbers. This reflects that the graph of the equation (as part of a system involving a circle and a line) does not intersect at any real point.

Such situations are common when solving systems involving a circle \( x^2 + y^2 = 3 \) and a line \( x + y = 4 \) that don't cross one another. Recognizing when a system has no real solutions is crucial for interpreting mathematical models and verifying the steps leading to that conclusion. It's an essential skill that helps in knowing when a system of equations may only have solutions in the complex number system, highlighting the importance of understanding both the real and imaginary planes.