if the roots of ax^2+bx+c=0 are real, rational, and equal, what is true about the graph of y= ax^2+bx+c?
1.) it intersects the x-axis at two distinct points.
2.) it lies entirely below the x-axis
3.) it lies entirely above the x-axis
4.) it is tangent to the x-axis
i think the answer is choice 1 but i'm not quite sure. can you please help me?
no
for 1.) the roots would have to be real, but different
The graph will intersect at the roots
since the roots are equal, the graph will have two equal intersection points, or in other words, it will just touch the x-axis.
so it must be tangent to the x-axis, thus 4)
Thank you once again!!! :)
Well, I'm here to help and bring some humor to the situation! The correct answer is actually choice number 4.) it is tangent to the x-axis. Think of it this way: if the roots of the quadratic equation are real, rational, and equal, it means that the graph will touch the x-axis at just one point, like a gentle clown nose boop. So, the graph of y = ax^2 + bx + c will be like a clown balancing perfectly on one foot, just grazing the x-axis!
You are correct! If the roots of the quadratic equation ax^2+bx+c=0 are real, rational, and equal, then the discriminant b^2-4ac=0. In such a case, the graph of y=ax^2+bx+c is tangent to the x-axis at the single root. Therefore, the correct answer is choice 4, "it is tangent to the x-axis."
To determine the correct answer, let's consider the nature of the roots of the quadratic equation ax^2 + bx + c = 0 when they are real, rational, and equal.
In general, a quadratic equation can have three types of roots:
1. Real, unequal roots: The quadratic equation intersects the x-axis at two distinct points.
2. Real, equal roots: The quadratic equation intersects the x-axis at one point, which means it is tangent to the x-axis.
3. Complex conjugate roots: The quadratic equation does not intersect the x-axis, meaning it lies entirely above or below the x-axis.
Since we know that the roots of this equation are real, rational, and equal, it falls into the second case - the quadratic equation is tangent to the x-axis.
Therefore, the correct answer is:
4.) It is tangent to the x-axis.