Solving of Quadratic Equations, Nature of Roots, Formulas, Applications and Examples

Quadratic Equations

Quadratic Equations have been around for centuries! Indian mathematicians Brahmagupta and Bhaskara II made some significant contributions to the field of quadratic equations. Although quadratic equations look complicated and generally strike fear among students, with a systematic approach they are easy to understand. In this Article You will find Solving of Quadratic Equations, Nature of Roots, Applications of Quadratic Roots, Quadratic Equation Formulas, Roots of Quadratic Equation, General Form of Quadratic Equation, Maximum and Minimum Value, Theory of Equations, Location of Roots, and Logarithmic Inequalities.

The general form of a quadratic equation in x is , a x² + bx + c = 0 , where a , b , c ∈ R & a ≠ 0.

• The solution of the quadratic equation , ax² + bx + c = 0 is given by x = \(\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\) The expression b2 – 4ac = D is called the discriminant of the quadratic equation.
• If α & β are the roots of the quadratic equation ax² + bx + c = 0, then;
  (i) α + β = – b/a
  (ii) α β = c/a
  (iii) α – β = \(\frac {\sqrt {D}}{a}\)
• Nature Of Roots:
  (A) Consider the quadratic equation ax² + bx + c = 0 where a, b, c ∈ R & a≠ 0 then
  (i) D > 0 ⇔ roots are real & distinct (unequal).
  (ii) D = 0 ⇔ roots are real & coincident (equal).
  (iii) D < 0 ⇔ roots are imaginary.
  (iv) If p + i q is one root ofa quadratic equation, then the other must be the conjugate p − i q & vice versa. (p , q ∈ R & i = \(\sqrt {-1}\))
  (B) Consider the quadratic equation ax2 + bx + c = 0 where a, b, c ∈ Q & a ≠ 0 then;
  (i) If D > 0 & is a perfect square , then roots are rational & unequal.
  (ii) If α = p + \(\sqrt {q}\) is one root in this case, (where p is rational & q is a surd) then the other root must be the conjugate of it i.e. β = p − \(\sqrt {q}\) & vice versa.
• A quadratic equation whose roots are α & β is (x − α)(x − β) = 0 i.e.
  x² − (α + β) x + α β = 0 i.e. x² − (sum of roots) x + product of roots = 0.
• Remember that a quadratic equation cannot have three different roots & if it has, it becomes an identity.
• Consider the quadratic expression , y = ax² + bx + c , a ≠ 0 & a , b , c ∈ R then
  (i) The graph between x , y is always a parabola . If a > 0 then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.
  (ii) ∀ x ∈ R , y > 0 only if a > 0 & b² − 4ac < 0 (figure 3) .
  (iii) ∀ x ∈ R , y < 0 only if a < 0 & b² − 4ac < 0 (figure 6) .
  Carefully go through the 6 different shapes of the parabola given below.
  
• Solution Of Quadratic Inequalities:
  ax² + bx + c > 0 (a ≠ 0).
  (i) If D > 0, then the equation ax² + bx + c = 0 has two different roots x₁ < x₂.
  Then a > 0 ⇒ x ∈ (−∞, x₁) ∪ (x₂, ∞)
  a < 0 ⇒ x ∈ (x₁, x₂)
  (ii) If D = 0, then roots are equal, i.e. x₁ = x₂.
  In that case a > 0 ⇒ x ∈ (−∞, x₁) ∪ (x₁, ∞)
  a < 0 ⇒ x ∈ φ
  (iii)Inequalities of the form \(\frac {P(x)}{Q(x)}\) 0 can be quickly solved using the method of intervals.
• Maximum & Minimum Value of y = ax² + bx + c occurs at x = − (b/2a) according as ; a < 0 or a > 0.\(\text { y } \in\left[\frac{4 \mathrm{ac}-\mathrm{b}^{2}}{4 \mathrm{a}}, \infty\right) \text { if } \mathrm{a}>0 \& \text { y } \in\left(-\infty, \frac{4 \mathrm{ac}-\mathrm{b}^{2}}{4 \mathrm{a}}\right] \text { if } \mathrm{a}<0\)
• Common Roots of 2 Quadratic Equations [Only One Common Root]:
  Let α be the common root of ax² + bx + c = 0 & a′x2 + b′x + c′ = 0 Thereforea α² + bα + c = 0 ; a′α² + b′α + c′ = 0. By Cramer’s Rule \(\frac{\alpha^{2}}{b c^{\prime}-b^{\prime} c}=\frac{\alpha}{a^{\prime} c-a c^{\prime}}=\frac{1}{a b^{\prime}-a^{\prime} b}\)  Therefore, α = \(\frac{c a^{\prime}-c^{\prime} a}{a b^{\prime}-a^{\prime} b}=\frac{b c^{\prime}-b^{\prime} c}{a^{\prime} c-a c^{\prime}}\). So the condition for a common root is (ca′ − c′a)² = (ab′ − a′b)(bc′ − b′c).
• The condition that a quadratic function
  f(x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c may be resolved into two linear factors is that;  abc + 2 fgh − af2 − bg2 − ch2 = 0 OR
  \(\left|\begin{matrix} a &h &g \\ h &b &f \\ g &f &c \end{matrix}\right| = 0\)
• Theory of Equations: If α₁, α₂, α₃, ……α_n are the roots of the equation;
  f(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 + …. + a_n-1x + a_n = 0 where a₀, a₁, …. a_n are all real & a₀ ≠ 0 then, \(\sum\alpha_{1}=-\frac{\mathrm{a}_{1}}{\mathrm{a}_{0}}\), \(\sum \alpha_{1} \alpha_{2}=+\frac{\mathrm{a}_{2}}{\mathrm{a}_{0}}, \sum \alpha_{1} \alpha_{2} \alpha_{3}=-\frac{\mathrm{a}_{3}}{\mathrm{a}_{0}}, \ldots \ldots, \alpha_{1} \alpha_{2} \alpha_{3}\ldots \ldots . \alpha_{\mathrm{n}}=(-1)^{\mathrm{n}} \frac{\mathrm{a}_{\mathrm{n}}}{\mathrm{a}_{0}}\)
  Note:
  (i) If α is a root of the equation f(x) = 0, then the polynomial f(x) is exactly divisible by (x − α) or (x − α) is a factor of f(x) and conversely .
  (ii) Every equation of nth degree (n ≥ 1) has exactly n roots & if the equation has more than n roots, it is an identity.
  (iii) If the coefficients of the equation f(x) = 0 are all real and α + iβ is its root, then α − iβ is also a root. i.e. imaginary roots occur in conjugate pairs.
  (iv) If the coefficients in the equation are all rational & α +\(\sqrt{\beta}\) is one of its roots, then α − \(\sqrt{\beta}\) is also a root where α, β ∈ Q & β is not a perfect square.
  (v) If there be any two real numbers ‘a’ & ‘b’ such that f(a) & f(b) are of opposite signs, then f(x) = 0 must have atleast one real root between ‘a’ and ‘b’ .
  (vi) Every eqtion f(x) = 0 of degree odd has atleast one real root of a sign opposite to that of its last term.
• Location Of Roots: Let f(x) = ax² + bx + c, where a > 0 & a, b, c ∈ R.
  (i) Conditions for both the roots of f (x) = 0 to be greater than a specified number ‘d’ are b² − 4ac ≥ 0; f(d) > 0 & (− b/2a) > d.
  (ii) Conditions for both roots of f(x) = 0 to lie on either side of the number ‘d’ (in other words the number ‘d’ lies between the roots of f(x) = 0) is f(d) < 0.
  (iii) Conditions for exactly one root of f(x) = 0 to lie in the interval (d , e) i.e. d < x < e are b²− 4ac > 0 & f(d) . f(e) < 0.
  (iv) Conditions that both roots of f(x) = 0 to be confined between the numbers p & q are (p < q). b2 − 4ac ≥ 0; f(p) > 0; f(q) > 0 & p < (− b/2a) < q.
• Logarithmic Inequalities
  (i) For a > 1 the inequality 0 < x < y & log_a x < log_a y are equivalent.
  (ii) For 0 < a < 1 the inequality 0 < x < y & log_a x > log_a y are equivalent.
  (iii) If a > 1 then log_a x < p ⇒ 0 < x < a^p
  (iv) If a > 1 then log_a x > p ⇒ x > a^p
  (v) If 0 < a < 1 then log_a x < p ⇒ x > a^p
  (vi) If 0 < a < 1 then log_a x > p ⇒ 0 < x < a^p