a system of equations consisting of a linear equation and a quadratic equation ______ has two solution
Always, sometimes, or never
the solutions given by the quadratic formula are ______ irrational
Always, Sometimes, or never
no solutions: x^2, x-1
one solution: x^2, x - 1/4
two solutions: x^2, x
Sometimes.
A system of equations consisting of a linear equation and a quadratic equation can have two solutions under certain conditions.
If the quadratic equation has two distinct real roots and the linear equation intersects with the graph of the quadratic equation at both of those points, then the system will have two solutions. However, if the quadratic equation has no real roots or if the linear equation does not intersect with the graph of the quadratic equation, then the system will not have two solutions.
A system of equations consisting of a linear equation and a quadratic equation can have either two solutions, one solution, or no solutions, depending on the specific equations involved.
To determine whether the system has two solutions, we can use the concept of the discriminant in quadratic equations. The discriminant is the expression that appears under the square root sign in the quadratic formula, which is used to solve quadratic equations.
If the discriminant is positive, the quadratic equation has two distinct real solutions. If it is zero, the quadratic equation has one real solution (a double root). And if the discriminant is negative, the quadratic equation has no real solutions (it has two complex conjugate solutions).
Now, in a system of equations, if the linear and quadratic equations are such that the linear equation and the associated quadratic equation have different discriminant values, then the system will have two solutions. This occurs when the linear equation intersects the quadratic equation at two distinct points or intersects the parabola of the quadratic equation twice.
Alternatively, if the discriminant values of the linear and quadratic equations are the same, the system can have one solution if the linear equation and the quadratic equation either intersect at one point or coincide. In this case, the discriminant of the quadratic equation will be zero.
Lastly, if the discriminant values of the linear and quadratic equations are different, and the quadratic equation has a negative discriminant (indicating no real solutions), the system will have no solutions.
Therefore, to determine if a system of equations consisting of a linear equation and a quadratic equation always, sometimes, or never has two solutions, you need to analyze the discriminant values of the quadratic equation involved in the system.