Quadratic equation Study Guide

📖 Core Concepts Standard form: $ax^{2}+bx+c=0$ with $a\neq0$ (quadratic, linear, constant coefficients). Root / zero / solution: Any $x$ satisfying the equation; a quadratic has exactly two roots counting multiplicity. Discriminant: $D=b^{2}-4ac$ decides how many / what type of real roots exist. Vertex: $x{\text{vertex}}=-\dfrac{b}{2a}$; the parabola’s highest/lowest point. Vieta’s relations: For roots $r{1},r{2}$, \[ r{1}+r{2}=-\frac{b}{a},\qquad r{1}r{2}=\frac{c}{a}. \] --- 📌 Must Remember Quadratic formula: $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Discriminant cases: $D>0$ → two distinct real roots. $D=0$ → one double (repeated) real root. $D<0$ → two complex‑conjugate roots. Zero‑Factor Property: $(px+q)(rx+s)=0 \;\Rightarrow\; px+q=0$ or $rx+s=0$. Vertex $x$‑coordinate: $-\dfrac{b}{2a}$. Reduced (monic) form: Divide by $a$ → $x^{2}+px+q=0$, $p=\dfrac{b}{a}$, $q=\dfrac{c}{a}$. --- 🔄 Key Processes Factoring by inspection (monic case): Find $q,s$ with $q+s=b$, $qs=c$. Write $(x+q)(x+s)=0$, solve each factor. Completing the square (general $a$): Divide by $a$: $x^{2}+\frac{b}{a}x+\frac{c}{a}=0$. Move constant: $x^{2}+\frac{b}{a}x=-\frac{c}{a}$. Add $\big(\frac{b}{2a}\big)^{2}$: $\big(x+\frac{b}{2a}\big)^{2}= \frac{b^{2}-4ac}{4a^{2}}$. Square‑root → $x+\frac{b}{2a}= \pm\frac{\sqrt{D}}{2a}$. Isolate $x$. Stable numeric root computation (avoid cancellation): Compute the larger root \[ R=\frac{-b+\operatorname{sgn}(b)\sqrt{D}}{2a}. \] Obtain the smaller root via Vieta: $r=\dfrac{c}{aR}$. --- 🔍 Key Comparisons Factoring vs. Quadratic formula Factoring: fast when integer roots exist; requires guessing $q,s$. Formula: works for any coefficients; may suffer numeric cancellation. Standard vs. Reduced form Standard: $ax^{2}+bx+c=0$ (coefficients as given). Reduced (monic): $x^{2}+px+q=0$ after dividing by $a$; simplifies the formula. Discriminant $D$ vs. Vertex $x$‑coordinate $D$ tells how many real roots. $x{\text{vertex}}$ tells where the parabola attains its extremum (midpoint of roots). --- ⚠️ Common Misunderstandings “If $a=0$ the equation is still quadratic.” – No, it becomes linear; $a$ must be non‑zero. “A discriminant of zero means no solution.” – It means one double real root, not “no” solution. “Complex roots are “imaginary” only when $b$ is negative.” – The sign of $b$ is irrelevant; $D<0$ guarantees a complex conjugate pair. “Subtracting $c/a$ in completing the square changes the equation.” – It’s moved to the other side, preserving equality. --- 🧠 Mental Models / Intuition Parabola symmetry: The axis $x=-\dfrac{b}{2a}$ is the midpoint between the two (real) roots – think of a seesaw balanced at the vertex. Vieta’s shortcut: Imagine the two roots as numbers whose sum and product are fixed by $-b/a$ and $c/a$; this lets you reconstruct one root from the other. Cancellation danger: When two numbers are almost equal, their difference loses digits – like subtracting two nearly identical prices; compute the bigger root first, then use $r{small}=c/(a\,r{big})$. --- 🚩 Exceptions & Edge Cases Zero discriminant: Polynomial can be written $a(x-r)^{2}$; both roots coincide at $r=-\dfrac{b}{2a}$. Very large/small coefficients: May cause overflow or underflow in $b^{2}$ or $4ac$; use scaled/reduced form or the stable root algorithm. Complex coefficients: Outline assumes real coefficients; with complex coefficients the discriminant test for “real vs. complex” does not apply. --- 📍 When to Use Which Factorable integer roots → Try Factoring first (quick mental check). Any coefficients, need exact answer → Use the Quadratic Formula (or reduced form). Need a quick estimate or avoid cancellation → Apply the stable numeric root method (large root first, then Vieta). Deriving vertex or axis of symmetry → Compute $x{\text{vertex}}=-\dfrac{b}{2a}$ directly. When $a\neq1$ and completing the square feels messy → Reduce to monic form first, then complete the square. --- 👀 Patterns to Recognize $b^{2}-4ac$ appearing → instantly think “discriminant → root type”. Coefficients satisfying $q+s=b$ and $qs=c$ → the quadratic is factorable. Parabola opening direction matches sign of $a$ (upward if $a>0$, downward if $a<0$). Double root ↔ discriminant zero ↔ graph touches the $x$‑axis at a single point. --- 🗂️ Exam Traps Choosing the wrong sign in the formula: Remember the “$\pm$” gives two roots; forgetting the “$-$” in $-b$ yields the opposite sign. Mixing up $b^{2}$ and $4ac$ when computing $D$: A common typo is $b^{2}+4ac$ → leads to an always‑positive discriminant. Using the standard formula with $a=0$: The denominator becomes zero; first check that the equation is truly quadratic. Cancelling before applying the formula: Simplifying $ax^{2}+bx+c$ by dividing by a common factor that isn’t $a$ can change the discriminant’s value if not done correctly. Assuming a real root exists because $b$ is positive: Real roots depend on $D$, not on the sign of $b$ alone. ---