A quadratic equation is an equation of the form ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0. Solving this equation involves the use of the discriminant, which is the expression b2 – 4ac. Knowing the sign of the discriminant helps to determine the number and type of solutions for the equation. When the discriminant is positive, the equation has two real and distinct solutions.

## Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree 2. It is typically written in the form ax2 + bx + c = 0, where a, b and c are real numbers and a ≠ 0. A quadratic equation can be solved using the quadratic formula, which is x = -b ± √b2 – 4ac/2a. The expression b2 – 4ac is known as the discriminant.

The sign of the discriminant determines the number and type of solutions for the equation. If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has two complex solutions.

## Graph of Positive Discriminant

The graph of a quadratic equation with a positive discriminant is a parabola that opens up. The parabola is symmetric about the y-axis and has two distinct real solutions. The graph of the equation y = x2 + 4x + 3 is an example of a quadratic equation with a positive discriminant. The graph looks like this:

The graph of a quadratic equation with a negative discriminant is a parabola that opens down. The parabola has two complex solutions and is not symmetric about the y-axis. The graph of the equation y = x2 – 4x + 3 is an example of a quadratic equation with a negative discriminant. The graph looks like this:

In conclusion,

For those learning algebra, it is important to understand the various types of quadratic equations and their accompanying graphs. A quadratic equation is a polynomial equation of the form ax2 + bx + c = 0, where a, b, and c are coefficients, and x is the unknown. The discriminant of a quadratic equation is a number that is derived from the coefficients of the equation and is used to determine the number and type of solutions, or roots, the equation has. A positive discriminant indicates the equation has two real, distinct solutions, or roots.

The graph of a quadratic equation that has a positive discriminant is a parabola, or U-shaped curve. The general shape of a graph of a quadratic equation is a parabola, regardless of the sign of the discriminant. However, a graph of a quadratic equation that has a positive discriminant opens upwards. The vertex of such parabola is the highest point on the parabola,representing the lowest minimum on the graph. The vertex of the parabola can be found by calculating the x-coordinate, referred to as the x-intercept, of the vertex.

The equation of a quadratic equation with a positive discriminant involves the coefficient a, which must be positive. Otherwise, the equation will describe a downward-pointing parabola, instead of an upward-pointing one. The sign of the coefficient a is necessary as it determines the shape of the graph. In other words, if the coefficient a of the equation is positive, then the equation will have a graph that is in the shape of a parabola that points upwards.

In conclusion, the graph of a quadratic equation with a positive discriminant is a U-shaped curve that points upwards. It is important to remember that the sign of the coefficient a must be positive in order for the graph to be in this shape. In addition, the vertex of a positive discriminant equation is the highest point on the parabola, representing the lowest minimum.