ax² + bx² + c = 0
a(x - p)² + q = 0
ax² + bx + c = 0, where a ≠ 0
(x - p)(x - q) = 0
To find the solutions of a quadratic equation.
To convert the quadratic into a linear equation.
To express the equation in standard form.
To determine the vertex of the parabola.
ax² + bx + c
b²-4ac
2a
(-b ± √(b²-4ac)) / (2a)
It does not determine the nature of solutions.
If > 0, two real and distinct solutions; if = 0, one real double root; if < 0, two complex solutions.
Only checks if roots are real or complex.
All discriminants lead to real solutions.
The axis of symmetry.
The values of a, b, and c.
The values of x that satisfy ax² + bx + c = 0.
The turning points of a quadratic graph.
Factoring, completing the square, quadratic formula.
Graphical representation only.
Using higher-order polynomial methods.
Trial and error only.
A line parallel to the x-axis.
The line x = b/(2a).
The line x = -b/(2a).
The horizontal line y = c.
Write the quadratic as a product of two binomials, set each to zero, solve for x.
Convert to linear form first, then solve.
Use the axis of symmetry to deduce x-values.
Only used for equations with no real solutions.
Used to transform any equation into a linear one.
Solve by converting function to vertex form x = y.
Rewrite as (x - p)² = q, solve by square root method.
Reverse factoring to find solutions.
Shows always only one solution at x = 0.
Shows where the parabola intersects the x-axis.
Always indicates three solutions possible.
Can solely determine solutions without constraints.
The highest or lowest point, located at (h, k).
Always at the origin in standard form.
The midpoint of roots regardless of axis of symmetry.
The same as the y-intercept of the quadratic.
By the sign of the coefficient a; > 0 up, < 0 down.
The slope of the axis of symmetry.
The value of c only.
The distance between roots.
Larger |a| compress, smaller |a| widen the parabola.
It doesn't affect the shape, only the position.
Only affects the y-intercept of the parabola.
Impact only lies in determining axis of symmetry.
When the leading term changes sign.
When the discriminant is less than zero.
When all terms of the quadratic cancel out.
In every quadratic solution to find complex coefficients.
Yes, if the discriminant is less than zero.
No, there are always at least two real roots.
No, discriminants positive or zero always ensure real roots.
Yes, but only when a and b are both zero.