Linear and Quadratic Systems

Front

What is a linear equation?

Back

An equation that makes a straight line when graphed. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What is a quadratic equation?

Back

An equation that can be expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What does it mean for a system of equations to have no solution?

Back

It means that the equations represent parallel lines that never intersect.

Front

What does it mean for a system of equations to have one solution?

Back

It means that the equations intersect at exactly one point.

Front

What does it mean for a system of equations to have two solutions?

Back

It means that the equations intersect at two distinct points, typically in the case of a linear equation intersecting a quadratic.

Front

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

Back

Set the linear equation equal to the quadratic equation and solve for x. Then substitute x back into either equation to find y.

Front

What is the vertex of a quadratic function?

Back

The highest or lowest point of the parabola represented by the quadratic function, found at the point (h, k) in the vertex form y = a(x-h)² + k.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is given by b² - 4ac and determines the nature of the roots: if positive, two real solutions; if zero, one real solution; if negative, no real solutions.

Front

What is the standard form of a linear equation?

Back

The standard form is Ax + By = C, where A, B, and C are integers, and A should be non-negative.

Front

What is the slope of a line?

Back

The slope (m) is a measure of the steepness of a line, calculated as the rise over run (change in y over change in x).