Solving quadratic equations

How the SAT tests this, with worked examples — plus the 2 skills it covers. Free to read, no account needed.

A quadratic has the form ax2+bx+c=0ax^2 + bx + c = 0, and there are three ways to solve one. Picking the right method is most of the speed.
et_quadratic_methods.png
Factoring is fastest when the numbers are friendly. Rewrite as a product equal to zero, then set each factor to zero.
Square rooting works when there is no middle term. From x2=16x^2 = 16 you get x=±4x = \pm 4, and forgetting the negative root is the most common error on the whole topic.
The quadratic formula always works
x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}
so use it whenever factoring is not obvious within a few seconds.
The expression under the root, b2−4acb^2 - 4ac, is the discriminant, and it tells you how many real solutions exist before you solve: positive gives two, zero gives one, negative gives none. Questions asking for a constant that makes an equation have exactly one solution are asking you to set it to zero.

Worked examples

Solve x2−5x+6=0x^2 - 5x + 6 = 0.
It factors neatly, so use that
(x−2)(x−3)=0  ⇒  x=2 or x=3{(x-2)(x-3) = 0 \;\Rightarrow\; x = 2 \text{ or } x = 3}
Solve 2x2=322x^2 = 32.
There is no middle term, so isolate and square root — keeping both signs
x2=16  ⇒  x=±4{x^2 = 16 \;\Rightarrow\; x = \pm 4}
How many real solutions does x2+3x+7=0x^2 + 3x + 7 = 0 have?
Compute the discriminant rather than solving
b2−4ac=9−28=−19{b^2 - 4ac = 9 - 28 = -19}
It is negative, so there are no real solutions.

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All the skills in Solving quadratic equations