Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula

Factoring, Completing the Square, and the Quadratic Formula

• Key goal: Solve quadratic equations of the form $ax^2+bx+c=0.$ If a factorization is possible, it is often the easiest route; otherwise use completing the square or the quadratic formula.

• When factoring is convenient:

  • If the leading coefficient is 1 (i.e., $a=1$), you look for two numbers that multiply to $c$ and add to $b$.
  • If such two numbers exist, you can factor the quadratic directly as
  • $(x + r)(x + s) = 0$ where $r+s=b$ and $rs=c.$
  • Example from the transcript: two numbers that sum to 5 and multiply to $-24$ are $8$ and $-3$, giving
    • $x^2+5x-24=(x+8)(x-3)=0.$
    • Roots: $x=-8\quad\text{or}\quad x=3.$
  • The coefficient in front of $x^2$ affects difficulty: if $a\neq 1$, directly factoring can be harder.
  • For a non-unit leading coefficient, one common approach is to factor out $a$ to get a monic quadratic inside, or use the AC method (see below).

• When the leading coefficient is not 1 (i.e., $a\neq 1$):

  • Factoring is often easier after transforming to a monic form or using the AC method.
  • AC method (factor by splitting middle term): find two numbers $m,n$ such that
  • $m\cdot n = a c$ and $m+n = b.$
  • Then rewrite the middle term $bx$ as $mx+nx$ and factor by grouping.

• Example discussion from the transcript about the $a=1$ case:

  • The equation $x^2+5x-24=0$ factors as $(x+8)(x-3)=0.$
  • Solutions: $x=-8$ and $x=3$.
  • Check: $(-8)\cdot 3=-24=c$ and $-8+3=-5$ which matches the sign convention with $b=5$ when expanded correctly as $x^2+5x-24.$

• Completing the square (alternative to factoring):

  • Start from$ax^2+bx+c=0.$ Divide by $a$ to get
  • $x^2+\frac{b}{a}x+\frac{c}{a}=0.$
  • To complete the square, add and subtract $\left(\frac{b}{2a}\right)^2$ inside the left-hand side:
  • $x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2 = -\frac{c}{a}+\left(\frac{b}{2a}\right)^2.$
  • The left side becomes a square:
  • $\left(x+\frac{b}{2a}\right)^2 = \frac{b^2-4ac}{4a^2} = \frac{D}{4a^2},$
    where $D = b^2-4ac$ is the discriminant.
  • Solve for $x$:
  • $x = -\frac{b}{2a} \pm \frac{\sqrt{D}}{2a}.$
  • This derivation leads to the quadratic formula; completing the square is another route to the same roots.

• The quadratic formula and the discriminant:

  • Discriminant: $D=b^2-4ac.$
  • Roots (the quadratic formula):
  • $x=\frac{-b\pm\sqrt{D}}{2a}.$
  • Interpretations by the sign of $D$:
  • If D>0 there are two distinct real roots.
  • If $D=0$ there is a repeated real root.
  • If D<0 there are two complex roots.

• Practical note on notation and practice:

  • In practice, you may not always explicitly write $a,b,c$ if you’re comfortable with the coefficients; you can work directly with the given quadratic. However, knowing $a,b,c$ helps you apply the discriminant and the standard formulas.
  • After solving, always check by expansion to verify the roots satisfy $ax^2+bx+c=0$.

• Word problem illustration (based on the transcript): word problems can yield a quadratic, especially with total quantities or times.

  • Example setup mentioned: one segment’s time is $s$, another is $s+100$, and you relate them via a total time expression.
  • General approach: set up unknowns (like $s$), express each quantity in terms of the unknowns, form an equation for the total (or another given constraint), and solve using factoring, completing the square, or the quadratic formula.

• Summary of strategies:

  • If factoring is possible and easy, factor and set each factor to zero to get the roots.
  • If a is not 1, try to make it monic or use AC method to factor.
  • If factoring is hard or not possible, use the quadratic formula with $D=b^2-4ac$.
  • Use completing the square to understand the derivation of the quadratic formula and for certain problem forms.
  • Always verify solutions in the original equation.