The quadratic formula
Some equations can't be untangled by moving things around - x^2 - 4x + 2 = 0
has an x squared and an x, and no amount of shuffling isolates it. The
quadratic formula is the master key for every equation shaped like
ax^2 + bx + c = 0:
x = (-b + sqrt(b^2 - 4*a*c)) / (2*a)
a = the number in front of x^2
b = the number in front of x
c = the constant on its own
(The full formula has a plus-or-minus before the square root - two roots. We
take the plus branch here: the larger root.)
The part under the square root, b^2 - 4*a*c, is called the discriminant -
if it comes out negative, the equation has no real solution, and sqrt of a
negative number is exactly where your calculator complains.
The starter shows the shape on x^2 - 3x + 2 = 0 (a=1, b=-3, c=2), where the
larger root is a clean 2. This lesson's equation doesn't come out clean - the
answer keeps its decimals, so the parentheses have to be exactly right.
Your task: find the larger root of x^2 - 4x + 2 = 0 (a=1, b=-4, c=2).
Careful with the signs: b is -4, so -b is +4.
You'll practice:
- Substituting into a formula with a square root and a fraction
- Handling a negative b (two sign flips that cancel - or don't, if you rush)