Linear / Quadratic Systems of Equations

Front

What is a linear equation?

Back

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. 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

A quadratic equation is a polynomial equation of degree two, typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

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. If they intersect at two points, there are two solutions; if they touch at one point, there is one solution; and if they do not intersect, there are no solutions.

Front

What is the substitution method in solving systems of equations?

Back

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation to find the values of the variables.

Front

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

Back

A system of equations has no solution when the equations represent parallel lines that never intersect.

Front

What is the vertex of a quadratic function?

Back

The vertex of a quadratic function is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. It can be found using the formula (-b/2a, f(-b/2a)).

Front

How do you graph a quadratic function?

Back

To graph a quadratic function, identify the vertex, axis of symmetry, and intercepts. Plot these points and draw a smooth curve through them.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, given by b^2 - 4ac. It 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 difference between a linear function and a quadratic function?

Back

A linear function has a constant rate of change and graphs as a straight line, while a quadratic function has a variable rate of change and graphs as a parabola.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants.