Quadratics flashcards

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What is the general form of a quadratic equation?

$ax^2 + bx + c = 0$, where $a, b, c$ are constants and $a \neq 0$.

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What is the quadratic formula for solving $ax^2 + bx + c = 0$?

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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What is the discriminant of a quadratic equation?

The discriminant, often denoted by $\Delta$, is the part of the quadratic formula under the square root: $\Delta = b^2 - 4ac$.

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What does it mean if the discriminant $b^2 - 4ac > 0$?

The equation has two distinct real roots. The graph intersects the x-axis at two different points.

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What does it mean if the discriminant $b^2 - 4ac = 0$?

The equation has one repeated real root (or two equal real roots). The graph touches the x-axis at a single point (the vertex is on the x-axis).

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What does it mean if the discriminant $b^2 - 4ac < 0$?

The equation has no real roots. The graph does not intersect the x-axis.

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What is the completed square form of a quadratic?

$a(x-h)^2 + k$. This form is also known as the vertex form.

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How do you find the vertex from the completed square form $y = a(x-h)^2 + k$?

The vertex is at the point $(h, k)$. Be careful with the sign of $h$.

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What is a common mistake when finding the vertex from $y = a(x+p)^2 + q$?

Incorrectly stating the x-coordinate of the vertex as $p$. The vertex is at $(-p, q)$ because the general form is $(x-h)^2$.

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When is the vertex of $y = ax^2 + bx + c$ a minimum point?

When the coefficient of $x^2$, which is $a$, is positive ($a > 0$). The parabola is U-shaped.

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When is the vertex of $y = ax^2 + bx + c$ a maximum point?

When the coefficient of $x^2$, which is $a$, is negative ($a < 0$). The parabola is n-shaped.

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What are the conditions for a quadratic curve to be 'always positive' (entirely above the x-axis)?

Two conditions must be met: 1. The parabola must be U-shaped ($a > 0$). 2. It must have no real roots ($b^2 - 4ac < 0$).