Quadratic Equations — JEE Mains Mathematics

Formulas (2)

• Sum, product and difference of roots · Sum, product, difference
  $$\alpha+\beta=-\frac{b}{a},\quad \alpha\beta=\frac{c}{a},\quad (\beta-\alpha)^2=\frac{b^2-4ac}{a^2}$$
• Beyond degree two: cubics and polynomial divisors · Vieta for a cubic
  $$\alpha+\beta+\gamma=-\frac{b}{a},\quad \sum\alpha\beta=\frac{c}{a},\quad \alpha\beta\gamma=-\frac{d}{a}$$

Watch out for (2)

• The difference needs its square
  → Sum, product and difference of roots
• The signs alternate
  → Beyond degree two: cubics and polynomial divisors

Formulas (2)

• Symmetric expressions from the sum and product · Squares to fourth powers
  $$\alpha^4+\beta^4=\left(\alpha^2+\beta^2\right)^2-2(\alpha\beta)^2$$
• High powers by a recurrence · Power-sum recurrence
  $$x^2=sx-p\ \Rightarrow\ P_n=sP_{n-1}-pP_{n-2}$$

Watch out for (2)

• Check that the roots can exist
  → Symmetric expressions from the sum and product
• Read the sign of the product
  → High powers by a recurrence

Formulas (2)

• A root shared by two equations · Condition for one common root
  $$(c_1a_2-c_2a_1)^2=(a_1b_2-a_2b_1)(b_1c_2-b_2c_1)$$
• Building an equation from its roots · Equation from its roots
  $$x^2-(r_1+r_2)x+r_1r_2=0$$

Watch out for (2)

• Proportional only for non-real roots
  → A root shared by two equations
• Divide by the leading coefficient
  → Building an equation from its roots

Formulas (3)

• The sign of the discriminant · Discriminant
  $$D=b^2-4ac,\qquad x=\frac{-b\pm\sqrt D}{2a}$$
• Rational and integer roots · Rational roots
  $$\sqrt{b^2-4ac}\in\mathbb{Z}\ \Rightarrow\ x=\frac{-b\pm\sqrt{D}}{2a}\ \text{is rational}$$
• Placing the roots on the number line · A number between the roots
  $$\alpha<k<\beta\quad\text{exactly when}\quad a\,f(k)<0$$

Watch out for (3)

• The x² coefficient can vanish
  → The sign of the discriminant
• Rational coefficients first
  → Rational and integer roots
• The vertex condition is not optional
  → Placing the roots on the number line

Formulas (3)

• Splitting at the critical points · Definition of modulus
  $$|x-a|=\begin{cases}x-a,& x\ge a\\ a-x,& x<a\end{cases}$$
• One modulus as the unknown · Equality in the triangle inequality
  $$|A+B|=|A|+|B|\quad\text{exactly when}\quad AB\ge0$$
• Greatest integer and fractional part · Greatest integer
  $$[x]=n\quad\text{exactly when}\quad n\le x<n+1$$

Watch out for (3)

• A root on the boundary is counted once
  → Splitting at the critical points
• Reject negative values of |x|
  → One modulus as the unknown
• The fractional part is below 1
  → Greatest integer and fractional part

Formulas (3)

• Exponential equations · Reciprocal pair
  $$t+\frac1t=k\ \Rightarrow\ t=\frac{k\pm\sqrt{k^2-4}}{2}$$
• Logarithmic equations · Swapped bases
  $$\log_b a=\frac{1}{\log_a b}$$
• A repeated expression as the unknown · Reciprocal substitution
  $$x^2+\frac1{x^2}=\left(x+\frac1x\right)^2-2$$

Watch out for (3)

• Every t must be positive
  → Exponential equations
• Check every base
  → Logarithmic equations
• Undo the substitution with its range
  → A repeated expression as the unknown