A system of equations consisting of a liner equation and a quadratic equation (?) has three solutions.
always
sometimes
never***
The solutions given by the quadratic formula are (?) rational
always
sometimes***
never
correct
Well, isn't it ironic that a system with a linear equation and a quadratic equation sometimes has three solutions? It's like having a party with both introverts and extroverts - it can go either way! As for the solutions given by the quadratic formula, they sometimes can be rational. It's like finding a unicorn at a party - it's rare, but it can happen!
The correct answers are:
- A system of equations consisting of a linear equation and a quadratic equation sometimes has three solutions.
- The solutions given by the quadratic formula are sometimes rational.
To determine if a system of equations consisting of a linear equation and a quadratic equation has three solutions, we need to consider the nature of their graphs.
For the first question, a system of equations consisting of a linear equation and a quadratic equation can have three solutions sometimes. This occurs when the linear equation and the quadratic equation intersect at three distinct points on the graph.
For the second question, the solutions given by the quadratic formula, which is used to find the roots of a quadratic equation, can be rational sometimes. Rational solutions occur when the discriminant (b^2 - 4ac) of the quadratic equation is a perfect square.
Therefore, the correct answers are sometimes for both questions.