If you have a general quadratic equation like this:

Then the quadratic formula will help you find the roots of a quadratic equation, i.e. the values of x where this equation is solved.

Simplify square roots

Factor and remove perfect squares:

(defn prime-factors
  ([n] (prime-factors 2 n))
  ([f n]
    (if (> n 1)
      (if (zero? (mod n f))
        (cons f (prime-factors f (/ n f)))
        (recur (inc f) n)))))

(defn perfect-squares [s]
  (loop [items (sort s) pairs []]
    (if (empty? items) pairs
      (if (= (first items) (second items))
        (recur (drop 2 items) (conj pairs (first items)))
        (recur (rest items) pairs)))))

(defn simplify-sqrt [sqrt]
  (let [sq (reduce * (perfect-squares (prime-factors sqrt)))]
    [sq (/ sqrt (* sq sq))]))

(defn quadratic-rational [[a b c]]
  (let [discriminant (simplify-sqrt (- (* b b) (* 4 a c)))]
    [(/ (- b) (first discriminant))
     (last discriminant) (/ (* 2 a) (first discriminant))]))

(quadratic-rational [3 24 48])

Graphing quadratic equations

Find the vertex and y-intercept

The x-coordinate of the vertex of a parabola in the form ax² + bx + c is -b/2a. The y coordinate can then be found by plugging the resulting value into the equation. The y-intercept is (0, c).

(defn graph [[a b c]]
  (let [vert-x (/ (- b) (* 2 a))
        vert-y (+ (* a (* vert-x vert-x))
          (* b vert-x)
          c)
        y-int [0 c]]
  {:vertex [vert-x vert-y]
   :y-int y-int}))

(graph [-1 14 0])

Now we have a vector [1 2 3] containing the values of a simplified rational expression in the form (1 +- sqrt2) over 3.

(defn quadratic-roots [[a b c]]
  (let [square (Math/sqrt (- (* b b) (* 4 a c)))]
  (str "x = {"
       (/ (+ (- b) square) (* 2 a))
       ", "
       (/ (- (- b) square) (* 2 a)) "}")))

[(quadratic-roots [-7 2 9]) (/ 9 7.0)]