Quadratic Functions and Equations: Chapter Review

Vocabulary and notation

Quadratic function

$f(x) = ax^2 + bx + c$ with $a \neq 0$; domain all reals, graph a parabola. With $a = 0$ it is linear and nothing here applies.

Parent parabola

$y = x^2$: even, vertex $(0, 0)$, range $[0, \infty)$, the shape every parabola is a scaled shift of.

Vertex, axis of symmetry

The turning point, and the vertical mirror through it. Inputs sharing an output are mirror partners, so the axis is their midpoint.

Zero, root, $x$-intercept

Three names for one set of numbers: where $f(x) = 0$, where the equation holds, where the graph meets the $x$-axis.

Repeated (double) root

A factor appearing twice, $f(x) = a(x - r)^2$: solution set $\{r\}$, and the parabola tangent to the $x$-axis at its vertex $(r, 0)$, touching without a sign change.

Factors over $\mathbb{Q}$, factors over $\mathbb{R}$

A factoring claim means nothing until a coefficient system is named: $x^2 - 2$ does not factor over $\mathbb{Q}$ but does over $\mathbb{R}$.

Discriminant $\Delta$

Standard notation for the number $b^2 - 4ac$ built from the coefficients of $ax^2 + bx + c$, $a \neq 0$.

Symmetric function of the roots

Unchanged when $r_1$ and $r_2$ are swapped, such as $r_1^2 + r_2^2$. Only these are fixed by the coefficients; $r_1 - r_2$ flips sign, so its sign is unrecoverable.

Formulas and theorems

• The three forms, and what each gives free

  $$\begin{gathered} ax^2 + bx + c \\ = a(x - h)^2 + k \\ = a(x - r_1)(x - r_2) \end{gathered}$$

  Use when $a \neq 0$, the same $a$ in all three. Standard gives $c = f(0)$ and the end behavior, arms up for $a > 0$, down for $a < 0$. Vertex gives $(h, k)$, axis $x = h$, and range $[k, \infty)$ or $(-\infty, k]$ by that same sign, inside signs reversed, so $x + 4$ means $h = -4$. Factored gives the zeros, and exists over $\mathbb{R}$ only when real zeros do.

• Axis and vertex from standard form

  $$x = -\frac{b}{2a}, \qquad y = c - \frac{b^2}{4a} = -\frac{\Delta}{4a}$$

  Use when $a \neq 0$. Coefficients keep their own signs, so a negative $b$ meets the formula's minus. Valid whether or not the parabola reaches the $x$-axis.

• Zero Product Property

  $$uv = 0 \iff u = 0 \ \text{ or } \ v = 0$$

  Use when Real numbers, and the other side must be exactly $0$: a product equal to any other number constrains neither factor.

• One zero buys one factor

  $$\begin{gathered} f(r) = 0 \\ \Longrightarrow \ ax^2 + bx + c \\ = (x - r)(ax + b + ar) \end{gathered}$$

  Use when $a \neq 0$ and $f(r) = 0$. So a quadratic factors over $\mathbb{R}$ exactly when it has a real zero, and over $\mathbb{Q}$ exactly when it has a rational zero.

  e.g. $3x^2 - 14x + 8$ at $r = 4$ gives $(x - 4)(3x - 2)$.

• Completing the square

  $$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$

  Use when For $ax^2 + bx + c$ with $a \neq 0$, factor $a$ out of the first two terms and complete the square inside, landing on $a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}$. Half of $b$ THEN squared, never negative, added and subtracted in the same line so the value is untouched. Unlike factoring, no step can fail.

• Square roots and the $\pm$

  $$(x + p)^2 = k \ \Longrightarrow \ x = -p \pm \sqrt{k}$$

  Use when Two real solutions when $k > 0$, one when $k = 0$, none over $\mathbb{R}$ when $k < 0$. $\sqrt{k}$ names only the nonnegative root, so the $\pm$ is the second solution, not decoration.

  e.g. $(x + 1)^2 = 9$: $x = 2$ or $x = -4$.

• Quadratic formula

  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = -\frac{b}{2a} \pm \frac{\sqrt{\Delta}}{2a}$$

  Use when $a \neq 0$, and $\Delta \ge 0$ for real roots. The $2a$ sits under the whole numerator. The right-hand split shows the roots as a mirror pair about the axis.

  e.g. $x^2 - 4x - 1 = 0$: $\Delta = 20$, so $x = 2 \pm \sqrt{5}$.

• The discriminant classifies the REAL roots

  $$\begin{gathered} \Delta > 0 \iff \text{two distinct real} \\ \Delta = 0 \iff \text{one repeated} \\ \Delta < 0 \iff \text{none real} \end{gathered}$$

  Text description

  Three parabolas on one axis: the first crosses it twice, the second touches it once, the third stays entirely above it.

  Use when Any real $a \neq 0$, $b$, $c$, rational or not; all three run both ways. $\Delta < 0$ denies REAL roots only. Equivalently $\Delta$ puts the vertex below, on, or above the $x$-axis.

  e.g. $3x^2 - 2x + 5$: $\Delta = -56$, so no real roots.

• Perfect-square test for factoring over $\mathbb{Q}$

  $$\begin{gathered} \text{factors over } \mathbb{Q} \\ \iff \Delta \ \text{is a perfect square} \end{gathered}$$

  Use when RATIONAL coefficients only; with irrational ones it says nothing. It demands strictly more than $\Delta > 0$: $3x^2 + 5x - 1$ has $\Delta = 37$, two real roots, no rational factorization, and "perfect square" means the square of a RATIONAL: $\tfrac12 x^2 + \tfrac12 x - 3$ has $\Delta = \tfrac{25}{4}$ and factors as $\tfrac12(x - 2)(x + 3)$.

  e.g. $2x^2 + 7x + 3$: $\Delta = 25$, so $(2x + 1)(x + 3)$.

• Sum and product of the roots, and what they build

  $$\begin{gathered} r_1 + r_2 = -\frac{b}{a}, \quad r_1 r_2 = \frac{c}{a} \\ r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2 r_1 r_2 \\ (r_1 - r_2)^2 = \frac{\Delta}{a^2} \end{gathered}$$

  Use when Any $a \neq 0$: the first two hold when $\Delta \ge 0$ and for the non-real pair too. Only the sum takes the extra minus, and both divide by $a$. The MONIC quadratic with these roots is $x^2 - (r_1 + r_2)x + r_1 r_2$, fixed by them only up to a nonzero scalar. A real pair with sum $s$ and product $p$ exists exactly when $s^2 - 4p \ge 0$; $\frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}$ needs $c \neq 0$.

  e.g. $3x^2 + 7x - 6$: sum $-\tfrac{7}{3}$, product $-2$.

• Pinning the leading coefficient

  $$\begin{gathered} y = a(x - r_1)(x - r_2) \\ \text{or} \ \ y = a(x - h)^2 + k \end{gathered}$$

  Use when The zeros, or the vertex, fix every parabola through them except its $a$, so one further point is needed; substitute it and solve the linear equation in $a$. That point must not be a zero or the vertex itself, which collapse to $0 = 0$ and determine nothing.

  e.g. Vertex $(3, -4)$ through $(0, 5)$: $5 = 9a - 4$, so $a = 1$.

• Where a quadratic is positive or negative

  $$\begin{gathered} a > 0: \\ f < 0 \ \text{strictly between the roots} \\ f > 0 \ \text{outside them} \end{gathered}$$

  Text description

  An upward parabola meeting the axis at two roots, negative only on the open interval between them and positive on both outer rays.

  Use when Two distinct real roots ($\Delta > 0$), everything on one side first; $a < 0$ reverses both. A quadratic changes sign only at a root, so a repeated root or no real root means no sign change at all.

  e.g. $(x - 3)(x + 2)$ is negative exactly on $(-2, 3)$.

Problem types, step by step

Exam traps

• Trap Distributing the factored-out $a$ to the square but not to the constant subtracted inside: $2\big((x + 3)^2 - 9\big) + 7$ finished as $2(x + 3)^2 - 2$.

  Fix That $-9$ is inside, so it doubles to $-18$: $2x^2 + 12x + 7 = 2(x + 3)^2 - 11$. Expand the finished form back.

• Trap Splitting a product equal to something other than zero: $(x - 1)(x - 2) = 6$ read as $x = 7$ or $x = 8$.

  Fix Neither works ($6 \cdot 5 = 30$); only $0$ pins a factor down. Expand, subtract $6$, refactor: $x = 4$ or $x = -1$.

• Trap Reading $\Delta \ge 0$ as "so it factors".

  Fix $x^2 - 2$ has $\Delta = 8$ and two real roots yet no rational factorization: that needs rational coefficients AND a perfect-square $\Delta$.

• Trap Reporting $\Delta < 0$ as "no roots", or as an automatically empty answer to an inequality.

  Fix Say no REAL roots. With $a > 0$, $\Delta < 0$ makes $f(x) < 0$ empty but $f(x) > 0$ all of $\mathbb{R}$: emptiness turns on the sign of $a$ and the symbol.

• Trap Collapsing an outside-the-roots answer into one interval, writing $[-2, 3]$ for $x^2 - x - 6 \ge 0$.

  Fix That is the exact complement. Two rays need a union: $(-\infty, -2] \cup [3, \infty)$.

• Trap Multiplying an inequality by $-1$ and keeping the symbol.

  Fix $-x^2 + 2x + 3 \ge 0$ becomes $x^2 - 2x - 3 \le 0$, never $\ge 0$. Reverse it, or read the downward parabola directly.

• Trap Treating a repeated root as a sign change, so $(x - 3)^2 > 0$ looks like two rays with a negative middle.

  Fix The square is positive everywhere but $3$, so the answer only punctures that point: $(-\infty, 3) \cup (3, \infty)$.

• Trap Writing the sum of the roots as $\frac{b}{a}$, or reading off $-b$ and $c$ when $a \neq 1$.

  Fix The sum is $-\frac{b}{a}$, the product $\frac{c}{a}$. For $2x^2 - 7x + 3$ the sum is $\tfrac{7}{2}$, not $7$.

• Trap Evaluating $r_1^2 + r_2^2$ as $(r_1 + r_2)^2$.

  Fix Subtract twice the product: sum $8$, product $2$ gives $64 - 4 = 60$, not $64$.