The Lumenary · Grades 9–12

Quadratic equations and functions

Name: ______________________

Date: ______________________

You will learn to

Solve quadratic equations by factorising, completing the square and the quadratic formula, and derive the formula
Use the discriminant $\Delta = b^2 - 4ac$ to decide how many real roots a quadratic has, and prove why
Relate the roots to the coefficients with Vieta's formulas $r + s = -b/a$ and $rs = c/a$
Sketch $y = ax^2 + bx + c$ with its vertex, axis of symmetry and intercepts
Solve optimisation problems by locating the vertex

1.
Compute the discriminant of $3x^2 - 5x + 2$ .
2.
Solve $x^2 - x - 12 = 0$ . Enter the solutions as a set, for example {1, 2}.
3.
How many different real roots does $4x^2 - 12x + 9$ have?
4.
Kofi solves $x^2 = 3x$ by dividing both sides by $x$ and gets $x = 3$ . What is wrong?
- A.Nothing: $x = 3$ is the only solution.
- B.Dividing by $x$ assumes $x \neq 0$ , so the solution $x = 0$ is lost.
- C.He should have taken square roots: $x = \pm\sqrt{3x}$ .
- D.The answer should be $x = -3$ .
5.
A goat pen is built against a straight wall using $60$ m of fencing for the other three sides. What is the largest possible area of the pen, in square metres?
6.
Without solving, find the sum and the product of the roots of $2x^2 - 7x + 3 = 0$ .
- A.sum $\tfrac72$ , product $\tfrac32$
- B.sum $-\tfrac72$ , product $\tfrac32$
- C.sum $7$ , product $3$
- D.sum $\tfrac32$ , product $\tfrac72$
7.
Find the positive number $k$ for which $x^2 + kx + 16 = 0$ has exactly one real root.
8.
In Mathlib, Real.sqrt of a negative number is defined to be 0. The theorem Lumenary.Quadratics.quadratic_formula has the hypothesis hΔ : 0 ≤ b ^ 2 - 4 * a * c. What would happen if we deleted it?
- A.Nothing: the statement stays true without it.
- B.The statement would become false, so no proof could exist.
- C.Lean would reject the statement because square roots of negatives do not exist.
- D.It would turn into the complex quadratic formula.
9.
Write real_roots(a, b, c) that returns the sorted list of the different real roots of $ax^2 + bx + c$ (with $a \ne 0$ ): two roots if $\Delta > 0$ , one if $\Delta = 0$ , and the empty list if $\Delta < 0$ . Use math.sqrt.
Computational — Practice

Answer key — Quadratic equations and functions

1.
$\Delta = (-5)^2 - 4 \cdot 3 \cdot 2 = 25 - 24 = 1 > 0$ , so there are two real roots: $x = \tfrac{5 \pm 1}{6}$ , that is, $1$ and $\tfrac23$ .
2.
$x^2 - x - 12 = (x - 4)(x + 3)$ , so by the zero-product property $x = 4$ or $x = -3$ . Check: $16 - 4 - 12 = 0$ and $9 + 3 - 12 = 0$ .
3.
$\Delta = (-12)^2 - 4 \cdot 4 \cdot 9 = 0$ , so there is exactly one real root, $x = \tfrac{12}{8} = \tfrac32$ . Indeed $4x^2 - 12x + 9 = (2x - 3)^2$ .
4.
Dividing by $x$ is only valid if $x \neq 0$ , so that case must be checked separately — and $x = 0$ does satisfy the equation. Safer: $x^2 - 3x = 0$ , so $x(x - 3) = 0$ , so $x = 0$ or $x = 3$ .
5.
$A(x) = x(60 - 2x) = -2x^2 + 60x$ has its vertex at $x = \tfrac{60}{4} = 15$ , giving $A(15) = 15 \cdot 30 = 450\ \text{m}^2$ . Completing the square: $A(x) = 450 - 2(x - 15)^2 \le 450$ .
6.
$r + s = -\tfrac{-7}{2} = \tfrac72$ and $rs = \tfrac{3}{2}$ . Check: the roots are $3$ and $\tfrac12$ , with sum $\tfrac72$ and product $\tfrac32$ .
7.
We need $\Delta = k^2 - 64 = 0$ , so $k = \pm 8$ ; the positive value is $k = 8$ . Then $x^2 + 8x + 16 = (x + 4)^2$ , with the single root $x = -4$ .
8.
Without hΔ the statement would be false. For $x^2 + x + 1$ we have Real.sqrt (-3) = 0, so the right-hand side says $x = -\tfrac12$ , while the left-hand side ( $x^2 + x + 1 = 0$ ) never holds for real $x$ . Lean's choice to make Real.sqrt total is convenient, but it means every theorem about square roots must state its hypotheses carefully.

import math

def real_roots(a, b, c):
    d = b * b - 4 * a * c
    if d < 0:
        return []
    if d == 0:
        return [-b / (2 * a)]
    s = math.sqrt(d)
    return sorted([(-b - s) / (2 * a), (-b + s) / (2 * a)])

The three branches are the three cases of the discriminant theorem. (Testing d == 0 with floating-point numbers is fine for these whole-number coefficients, but for measured data you would allow a small tolerance.)