Let be a quadratic function whose graph is a parabola opening upwards
MATHEMATICS · Multi-part · AoPS · Mathematics · Quadratic Equations - Part 2 · Quadratic Equations - Part 2
Problem
Let y=x2+bx+c be a quadratic function whose graph is a parabola opening upwards. The parabola touches (is tangent to) the x–axis at the point (4,0).
(i) Write the condition that must be satisfied by the coefficients b and c so that the equation x2+bx+c=0 has exactly one real root.
(ii) Using the information given, determine the values of b and c.
(iii) Verify your answer by calculating the discriminant and, on the basis of that value, explain why the graph is tangent to the x–axis.
(iv) Draw a neat labelled sketch of the parabola, showing the point of tangency and the axis of symmetry.