Lesson 4 — Quadratic function

Solve $x^2 - 5x + 6 = 0$.

Solve $2x^2 + 5x - 3 = 0$.

Solve $4x^2 - 12x + 9 = 0$.

Check whether $x^2 + x + 1 = 0$ has real roots.

Solve $x^2 - 7x + 10 = 0$.

Solve $x^2 + 2x - 4 = 0$.

Determine the values of $k$ for which $x^2 + 2x + k = 0$ has two distinct real roots.

Determine $k$ such that $f(x) = x^2 + kx + 9$ has a double root.

Prove Vieta's relations: if $x_1, x_2$ are roots of $ax^2 + bx + c = 0$, then $x_1 + x_2 = -b/a$ and $x_1 x_2 = c/a$.

Prove the quadratic formula by completing the square.

Determine the vertex of $f(x) = x^2 - 4x + 3$ and classify it.

Rewrite $f(x) = 2x^2 - 8x + 5$ in canonical form $a(x - x_V)^2 + y_V$.

Determine the vertex of $g(x) = -3x^2 + 4x - 1$ and rewrite in canonical form.

Find the quadratic with roots $-2$ and $5$ that passes through $(0, -10)$.

Find the quadratic with vertex at $(1, -3)$ that passes through $(3, 5)$.

For $y = a(x-3)^2 + 5$ with $a \neq 0$: what is the vertex, and how does $a$ affect the graph?

Use Vieta's relations to solve $x^2 - x - 2 = 0$ without applying the quadratic formula.

Prove that the x-coordinate of the vertex is the arithmetic mean of the roots (when they exist).

Determine $m$ for which $f(x) = mx^2 + (m+1)x + 1$ has vertex on the $y$-axis.

For $f(x) = x^2 - 2x - 8$: determine roots, vertex and canonical form.

Solve $x^2 - x - 6 < 0$.

Solve $x^2 \geq 9$.

Solve $-x^2 + 4x + 5 \geq 0$.

For which values of $b$ is the function $f(x) = 2x^2 + bx + 8$ positive for all $x \in \mathbb{R}$?

Solve $4x^2 - 3x + 4 > 0$.

Construct the sign table of $f(x) = x^2 - 3x - 10$ and indicate the intervals where $f < 0$ and where $f \geq 0$.

For which values of $m$ does the equation $x^2 + mx + (m + 3) = 0$ have two distinct real roots?

Prove, using canonical form, that for $a > 0$ the minimum value of $f(x) = ax^2 + bx + c$ is $y_V = -\Delta/(4a)$, attained at $x = x_V$.

Projectile launched vertically: $h(t) = 30t - 5t^2$ (m, s). (a) Time and maximum height? (b) When does it return to ground?

Two positive numbers whose sum is 100. Which maximize the product?

60 m of fencing covering 3 sides of a rectangle (wall on the fourth side). What dimensions maximize the area?

Cost $C(q) = q^2 - 30q + 250$ (R$) for $q$ units. Which $q$ minimizes the cost? What is the minimum cost?

Revenue $R(p) = p(200 - 4p)$. (a) Zeros? (b) Price for maximum revenue? (c) Maximum revenue?

A rectangular garden has length 4 m greater than the width and area 96 m². What are the dimensions?

Revenue $R(q) = 200q$ and cost $C(q) = 2q^2 + 30q + 200$. (a) Profit $L(q)$? (b) $q$ that maximizes it? (c) Maximum profit?

Ball trajectory: $h(d) = -0.1d^2 + d + 1$ m, where $d$ is horizontal distance. (a) Maximum height and where? (b) Where does it hit the ground?

Shop sells 100 units/week at R$ 50.00. Each R$ 1.00 reduction brings 5 extra customers. What price maximizes weekly revenue?

Rectangular chicken coop with 300 m of fencing, divided in half by a fence parallel to one side. What dimensions maximize total area?

Prove that, given $x + y = S$ with $x, y > 0$, the product $xy$ is maximized when $x = y = S/2$.

Prove that, when $\Delta \geq 0$, the quadratic function $f(x) = ax^2 + bx + c$ can be written as $f(x) = a(x - x_1)(x - x_2)$.

For which values of $p, q$ is the function $f(x) = x^2 + 2px + q$ non-negative for all $x \in \mathbb{R}$?

Prove the sign theorem of the quadratic: with $a > 0$ and $\Delta > 0$ (roots $x_1 < x_2$), $f(x) < 0$ if and only if $x \in (x_1, x_2)$.