Nonlinear Systems of Equations

Front

What is a nonlinear system of equations?

Back

A nonlinear system of equations is a set of equations in which at least one equation is nonlinear, meaning it cannot be expressed as a straight line when graphed.

Front

What is the general form of a quadratic equation?

Back

The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

How do you find the intersection of a linear and a quadratic equation?

Back

To find the intersection, set the equations equal to each other and solve for the variable. The solutions will give the points of intersection.

Front

What does it mean if a system of equations has no solutions?

Back

A system has no solutions if the equations represent parallel lines or curves that do not intersect.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots: if positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.

Front

What is the vertex of a parabola?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on its orientation (opening upwards or downwards).

Front

How can you determine the number of solutions in a linear-quadratic system?

Back

The number of solutions can be determined by analyzing the graphs of the equations: they can intersect at 0, 1, or 2 points.

Front

What is the standard form of a quadratic function?

Back

The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex.

Front

What is the role of the leading coefficient in a quadratic function?

Back

The leading coefficient (a) determines the direction of the parabola's opening: if a > 0, it opens upwards; if a < 0, it opens downwards.

Front

How do you solve a quadratic equation by factoring?

Back

To solve by factoring, express the quadratic in the form (x - p)(x - q) = 0, then set each factor to zero to find the solutions.