Number of Solutions for Quadratic Equations
Number of solutions for quadratic equations is a Grade 11 Algebra 1 concept from enVision Chapter 9 connecting graph behavior to algebraic solutions. A quadratic y = ax^2 + bx + c can have 0, 1, or 2 x-intercepts matching the 0, 1, or 2 real solutions. Two distinct real solutions mean the parabola crosses the x-axis twice — y = x^2 - 5x + 4 gives x = 1 and x = 4. One repeated solution means the vertex touches the axis — y = x^2 - 6x + 9 factors to (x-3)^2 = 0, giving x = 3. No real solutions means the parabola never touches the x-axis — y = x^2 + 5 has no real roots since x^2 = -5 is impossible.
Key Concepts
The $x$ intercepts of the graph of $y = ax^2 + bx + c$ are the solutions of $ax^2 + bx + c = 0$. There are three possibilities:.
1. If both solutions are real numbers, and unequal, the graph has two $x$ intercepts.
Common Questions
How many solutions can a quadratic equation have?
Zero, one, or two real solutions, corresponding to 0, 1, or 2 x-intercepts on the parabola.
Why does y = x^2 - 5x + 4 have two x-intercepts?
It factors to (x-1)(x-4) = 0, giving x = 1 and x = 4 — two distinct real solutions. The parabola crosses the x-axis at both points.
Why does y = x^2 - 6x + 9 have only one x-intercept?
It factors to (x-3)^2 = 0, giving one repeated solution x = 3. The vertex of the parabola sits exactly on the x-axis.
Why does y = x^2 + 5 have no x-intercepts?
Setting x^2 + 5 = 0 gives x^2 = -5, which has no real solutions. The parabola sits entirely above the x-axis.
How does the discriminant relate to the number of solutions?
Positive discriminant = 2 solutions, zero discriminant = 1 solution, negative discriminant = 0 real solutions.
What is a repeated solution in a quadratic?
When the discriminant equals zero, both solutions have the same value. The parabola is tangent to the x-axis at that point, which is also the vertex.