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In a quadratic equation of the standard form \( ax^2 + bx + c = 0 \), the numbers \( a \), \( b \), and \( c \) are called coefficients. These coefficients are essential for identifying the shape and position of the parabola represented by the quadratic equation.
In the given equation \( 2a^2 - 3a + 5 = 0 \), we have:

• \( a = 2 \)
• \( b = -3 \)
• \( c = 5 \)

Each coefficient plays a unique role:

• \( a \): Affects the curvature (how wide or narrow the parabola is). Since \( a = 2 \) is positive, the parabola opens upwards.
• \( b \): Influences the position of the vertex and axis of symmetry of the parabola.
• \( c \): Determines the y-intercept, which is the point where the parabola crosses the y-axis.

Understanding and identifying these coefficients make it easier to solve and graph the quadratic equation.

The discriminant is a value computed from the coefficients of a quadratic equation and is given by the formula \( D = b^2 - 4ac \).
This value helps us determine the nature of the solutions without actually solving the equation.
Let's calculate the discriminant for our equation \( 2a^2 - 3a + 5 = 0 \):

• \( a = 2 \)
• \( b = -3 \)
• \( c = 5 \)

Plugging in these values, we get:
\[ D = (-3)^2 - 4 \times 2 \times 5 = 9 - 40 = -31 \]
Here's what the discriminant tells us:

• If \( D > 0 \), there are two distinct real solutions.
• If \( D = 0 \), there is one real solution.
• If \( D < 0 \), there are two complex solutions.

In our case, \( D = -31 \) which is less than zero, meaning our quadratic equation has two complex solutions.

Complex solutions arise when the discriminant of a quadratic equation is less than zero ( \( D < 0 \)).
Complex numbers include a real part and an imaginary part and are usually written in the form \( a + bi \), where \( i \) is the imaginary unit, which satisfies \( i^2 = -1 \).
For our quadratic equation \( 2a^2 - 3a + 5 = 0 \), since we found \( D = -31 \), we know the solutions will be complex.
The quadratic formula, used to find the solutions, is given by:
\[ a = \frac{-b \pm \sqrt{ b^2 - 4ac }}{2a} \]
Since \( D = -31 \), we have:
\[ a = \frac{-(-3) \pm \sqrt{ -31 }}{2 \times 2} \]
\[ a = \frac{3 \pm \sqrt{ -31 }}{4} = \frac{3 \pm \sqrt{ 31} i }{4} \]
These solutions, \( \frac{3 \pm \sqrt{ 31} i }{4} \), are our two complex solutions. Remember:

• The \( \pm \) sign means we have two solutions.
• \(i \) denotes the imaginary unit.

Understanding complex solutions is crucial for solving many quadratic equations, especially those involving negative discriminants.