AlgebraAlgebra 2

Quadratic Equations Study Guide

A quadratic equation has the form $ax^2 + bx + c = 0$ with $a \ne 0$ . It has at most two solutions, and four standard methods find them.

Key concepts

Fastest when the quadratic factors over the integers: set each factor equal to zero.

When there is no $x$ -term, isolate $x^2$ and take both square roots.

Rewrites the quadratic as $a(x - h)^2 + k$ , which also gives the vertex.

Works for every quadratic; the discriminant $b^2 - 4ac$ tells you how many real solutions to expect.

Discriminant: $b^2 - 4ac$ : positive gives two real roots, zero gives one repeated root, negative gives two complex roots.
Vertex: The turning point of the parabola, at $x = -\frac{b}{2a}$ .
Root: An input that makes the expression equal zero.

Quadratic formula

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

Vertex x-coordinate

x = -\frac{b}{2a}

Sum and product of roots

r_1 + r_2 = -\frac{b}{a},\quad r_1 r_2 = \frac{c}{a}

1.Solve $x^2 - 7x + 12 = 0$ .
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$x = 3$ or $x = 4$
2.Solve $x^2 = 18$ .
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$x = \pm 3\sqrt{2}$
3.How many real roots does $x^2 + 2x + 5 = 0$ have?
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None (discriminant $-16$ )

x^2 - 5x + 6 = 0

A quadratic that factors over the integers.

2x^2 + 3x - 2 = 0

Leading coefficient other than one: the quadratic formula works every time.

x^2 = 49

Remember both square roots.

x^2 + 4x + 5 = 0

A negative discriminant gives complex roots.