Quadratic Equations and Functions - 1st Quarter Review

Quadratic Formula and Roots

• Quadratic Formula: roots of $ax^2+bx+c=0$ are
  $x=\frac{-b\,\pm\,\sqrt{b^2-4ac}}{2a}.$

• Examples (solve for x):

  • Solve $x^2-7x+10=0$.
  • Factorization: $x^2-7x+10=(x-5)(x-2)$
  • Roots: $x=5\quad ext{or}\quad x=2.$
  • Solve $2x^2+3x-9=0$.
  • Discriminant: $D=b^2-4ac=3^2-4(2)(-9)=9+72=81.$
  • Roots: $x=\frac{-3\pm\sqrt{81}}{2\cdot 2}=\frac{-3\pm9}{4}.$
  • So $x=\frac{6}{4}=\frac{3}{2}\quad\text{or}\quad x=\frac{-12}{4}=-3.$

• Sum and Product of the Roots

  • For roots $r<em>1,r</em>2$ of $ax^2+bx+c=0$,
  • Sum: $r1+r2=-\dfrac{b}{a}$
  • Product: $r1r2=\dfrac{c}{a}$
  • If roots are known, the equation can be written as
    $x^2-(r<em>1+r</em>2)x+(r<em>1r</em>2)=0.$

• Example: roots 8 and –5

  • Sum: $8+(-5)=3$
  • Product: $8\cdot(-5)=-40$
  • Equation in standard form: $x^2-3x-40=0.$

• Sum and Product without solving (practice)

  • Given equation $x^2-12x-3=0$ (rewriting $x^2-12x=3$ as $x^2-12x-3=0$),
  • Sum of roots: $12$
  • Product of roots: $-3$

• Discriminant and Nature of Roots

  • Discriminant: $D=b^2-4ac.$
  • If $D=0$: roots are real and equal.
  • If D>0 and $D$ is a perfect square: roots are real, rational, and unequal.
  • If D>0 and $D$ is not a perfect square: roots are real, irrational, and unequal.
  • If D<0: roots are imaginary (complex).

• Quick guide for nature of roots

  • $D=0\Rightarrow$ real & equal
  • D>0 and perfect square $\Rightarrow$ real, rational, unequal
  • D>0 and not perfect square $\Rightarrow$ real, irrational, unequal
  • D<0\Rightarrow imaginary

• Discriminant practice
  1) Find the discriminant of $x^2+4x+5=0$ and describe the roots.

  • D=4^2-4(1)(5)=16-20=-4<0
  • Roots are imaginary.
    2) Find the discriminant of $3x^2-2x-1=0$ and describe the roots.
  • $D=(-2)^2-4(3)(-1)=4+12=16$
  • D>0 and a perfect square ⇒ real, rational, unequal roots.

• Rational Algebraic Equations

  • 1) Solve: $\dfrac{x+2}{3}=5x.$
  • Multiply both sides by 3: $x+2=15x$
  • Bring terms together: $2=14x$
  • $x=\dfrac{1}{7}.$
  • 2) Solve: $\dfrac{8}{x}=\dfrac{x}{18}.$
  • Cross-multiply: $8\cdot18=x^2$
  • $x^2=144$
  • $x=\pm12.$ (Note: x=0 is not allowed in the original equation.)
  • 3) A rational-application word problem
  • Slower crew takes 8 days; faster crew takes 4 days to complete the same wall.
  • Rates: slower = $1/8$ per day; faster = $1/4$ per day.
  • If slower works alone for 3 days: work done = $3\cdot \tfrac{1}{8}=\tfrac{3}{8}$; remaining = $1-\tfrac{3}{8}=\tfrac{5}{8}$
  • Combined rate = $\tfrac{1}{8}+\tfrac{1}{4}=\tfrac{3}{8}$ per day.
  • Time to finish remaining $\tfrac{5}{8}$ at $\tfrac{3}{8}$ per day:
    $t=\dfrac{\tfrac{5}{8}}{\tfrac{3}{8}}=\dfrac{5}{3}\text{ days}.$
  • Therefore, time after joining: $\dfrac{5}{3}$ days; total time from start: $3+\dfrac{5}{3}=\dfrac{14}{3}$ days (about 4.67 days).
  • 4) Printing presses word problem
  • Slower press: completes order in 10 hours ⇒ rate $1/10$ per hour; Faster: 6 hours ⇒ $1/6$ per hour.
  • Slower works alone for 4 hours: amount completed = $4\cdot\tfrac{1}{10}=\tfrac{2}{5}$; remaining = $1-\tfrac{2}{5}=\tfrac{3}{5}.$
  • Joined rate = $\tfrac{1}{10}+\tfrac{1}{6}=\tfrac{8}{30}=\tfrac{4}{15}$ per hour.
  • Time to finish remaining: $t=\dfrac{\tfrac{3}{5}}{\tfrac{4}{15}}=\dfrac{3}{5}\cdot\dfrac{15}{4}=\dfrac{9}{4}=2.25\text{ hours}.$
  • Total time from start: $4+2.25=6.25\text{ hours}$; or remaining work finished in 2.25 hours after joining.

• Quadratic Inequality

  • 1) Solve x^2+4x-21<0.
  • Factor: (x+7)(x-3)<0.
  • Parabola opens upward; negative between roots: \(-7<x<3).
  • Solution: $(-7,\,3).$
  • 2) Solve $x^2-8x+15\ge 0.$
  • Factor: $(x-3)(x-5)\ge 0.$
  • Since parabola opens upward, inequality holds for $x\le 3\quad\text{or}\quad x\ge 5.$

Quadratic Functions and Graphs

• Standard form: $f(x)=ax^2+bx+c.$

• Vertex form: $f(x)=a(x-h)^2+k$ where the vertex is $(h,k).$

• Vertex coordinates from standard form

  • $h=-\frac{b}{2a}$
  • $k=\frac{4ac-b^2}{4a}$

• Vertex form provides direct access to the vertex and allows easy graphing and analysis of shifts, stretches, and vertical translations.

• Common features

  • Axis of symmetry: $x=h$
  • If a>0, parabola opens upward; the vertex is a minimum. If a<0, opens downward; the vertex is a maximum.
  • Range: depends on sign of $a$ and the vertex $k$; for example, if a>0, range is $[k,\infty)$.

• Example problem 1: Given $f(x)=2x^2-8x+11$

  • Vertex: $h=-\frac{-8}{2\cdot 2}=\frac{8}{4}=2$
  • Vertex value: $k=f(2)=2(4)-16+11=8-16+11=3$
  • Vertex form: $f(x)=2(x-2)^2+3$
  • Axis of symmetry: $x=2$
  • Min value: $3$; Range: $[3,\infty)$
  • Table of values (example):
  • For symmetry around x=2, evaluate at x=0,1,2,3,4:
  • $f(0)=11,\, f(1)=5,\, f(2)=3,\, f(3)=5,\, f(4)=11.$

• Graph: interpret visually (parabola opens upward, vertex at (2,3))

• EQUATION 1 (vertex form construction)

  • Task: Determine the equation of the graph in vertex form given a Vertex and a Point on the graph.
  • Method (general): If the vertex is $(h,k)$ and a point on the graph is $(x<em>1,y</em>1)$, then
  • The form is $f(x)=a(x-h)^2+k$ where
  • $a=\frac{y<em>1-k}{(x</em>1-h)^2}$
  • This yields the explicit vertex-form equation once a, h, k are known.

• Problem-Solving: Applications involving triangles, area, and projectile paths

  • 1) Rectangular garden
  • Length is 5 meters longer than width; area is 84 m^2.
  • Let width = w; length = w+5.
  • Equation: $w(w+5)=84\Rightarrow w^2+5w-84=0.$
  • Discriminant: $D=5^2-4(1)(-84)=25+336=361=19^2.$
  • Solutions: $w=\frac{-5\pm19}{2}$ ⇒ $w=7\text{ or }w=-12.$
  • Physical width: $w\ge 0$, so $w=7$, and length = $12$.
  • 2) Soccer ball trajectory
  • Model: $y=-2x^2+8x+6$ with $x$ in seconds, $y$ in feet.
  • Vertex: $x=-\dfrac{b}{2a}=-\dfrac{8}{2(-2)}=2$
  • Vertex value: $y=f(2)=-2(4)+16+6=14$
  • Axis of symmetry: $x=2$
  • Max value: $14$ (since a<0)
  • Range: $(-\infty, 14]$
  • Table of Values: compute around the vertex, e.g., at x = 0,1,2,3,4 to illustrate the shape.

• Connections and relevance

  • The discriminant connects to the nature of the roots and helps decide whether solving numerically is needed.
  • Sum/product relationships connect the roots to the coefficients, enabling construction of quadratic equations from given roots.
  • Vertex form provides a direct link to the graph: vertex location, symmetry, and extremum values, facilitating graphing and optimization tasks.
  • Real-world problems (area, motion) are naturally modeled by quadratic relationships; solving these translates algebra into practical answers.

• Practical and ethical considerations

  • When applying algebra to real-world problems, respect constraints (e.g., width must be positive in geometry problems).
  • Interpret the domain and physical feasibility of solutions; some mathematical results may be extraneous in real contexts (like negative dimensions).

• Summary of key formulas to memorize

  • Quadratic Formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
  • Discriminant: $D=b^2-4ac$
  • Sum of roots: $r<em>1+r</em>2=-\frac{b}{a}$
  • Product of roots: $r<em>1r</em>2=\frac{c}{a}$
  • Vertex: $h=-\frac{b}{2a},\quad k=\frac{4ac-b^2}{4a}$
  • Vertex form: $f(x)=a(x-h)^2+k$
  • Range indications from vertex and a: if a>0, range begins at $k$; if a<0, range ends at $k$.

• Quick reference table (selected results)

  • For $f(x)=2x^2-8x+11$:
  • Vertex: $(2,3)$; Vertex form: $f(x)=2(x-2)^2+3$
  • Axis: $x=2$; Min value: $3$; Range: $[3,\infty)$
  • For the discriminant problems:
  • x^2+4x+5=0\Rightarrow D=-4<0 (imaginary roots)
  • 3x^2-2x-1=0\Rightarrow D=16>0 (real, rational, unequal roots)

• Note on units and dimensions in word problems

  • Keep track of units (meters, seconds, inches, etc.) and ensure consistency when forming equations.

• Quick study tips

  • Practice deriving the sum/product relations from the quadratic coefficient structure.
  • Check cases for D to quickly classify roots without solving fully.
  • Use vertex form to quickly identify extremum and axis of symmetry when graphing.

• References to the material in this review

  • Quadratic Formula and roots; Sum and Product of Roots; Discriminant and Nature of Roots.
  • Rational Algebraic Equations; Quadratic Inequalities.
  • Quadratic Functions: standard form, vertex form, vertex calculations, and graph features.
  • Problem-solving applications and real-world modeling with quadratics.

• Final notes

  • Practice solving both algebraic and application problems to strengthen connections between formulas and real-world interpretations.