Quadratic-equations Formula Sheet

A quadratic equation is a second-degree polynomial equation of the form $ax^2+bx+c=0$, where It typically has two solutions, which can be real or complex depending on the discriminant $b^2-4ac$.

Neetesh Kumar | June 03, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Solution of Quadratic Equation & Relation Between Roots & Coefficients
• 2. Nature of Roots
• 3. Common Roots of Two Quadratic Equations
• 4. Roots Under Particular Cases
• 5. Maximum & Minimum Values of Quadratic Expression
• 6. Location of Roots
• 7. General Quadratic Expression in Two Variables
• 8. Theory of Equations

1. Solution of Quadratic Equation & Relation Between Roots & Coefficients:

(a) The solutions of the quadratic equation, $ax^2 + bx + c = 0$ is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

(b) The expression $b^2 - 4ac \equiv \Delta$ is called the discriminant of the quadratic equation.

(c) If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, then:

• $(i) \alpha + \beta = -\frac{b}{a}$
• $(ii) \alpha\beta = \frac{c}{a}$
• $(iii) |\alpha = \beta| = \frac{\sqrt{D}}{|a|}$

(d) Quadratic equation whose roots are $\alpha$ and $\beta$ is $(x - \alpha)(x - \beta) = 0$ i.e.
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$ i.e. $x^2 -$ (Sum of Roots)x + Product of Roots = 0

(e) If $\alpha$, $\beta$ are roots of equation $ax^2 + bx + c = 0$, we have identity in $x$ as $ax^2 + bx + c = a(x - \alpha)(x - \beta)$

2. Nature of Roots:

(a) Consider the quadratic equa tion $ax^2 + bx + c = 0$ where $a, b, c \in \mathbb{R}$ & $a \neq 0$ then:

• $(i) \Delta > 0 \Leftrightarrow$ roots are real & distinct (unequal).
• $(ii) \Delta = 0 \Leftrightarrow$ roots are real & coincident (equal).
• $(iii) \Delta < 0 \Leftrightarrow$ roots are imaginary.
• $(iv)$ If $p + iq$ is one root of a quadratic equation, then the other root must be the conjugate $p - iq$ & vice versa. $(p, q \in \mathbb{R}$ & $i = \sqrt{-1})$.

(b) Consider the quadratic equation $ax^2 + bx + c = 0$ where $a, b, c \in \mathbb{Q}$ & $a \neq 0$ then:

• $(i)$ If $\Delta$ is a perfect square, then roots are rational.
• $(ii)$ If $\alpha = p + \sqrt{q}$ is one root in this case, (where $p$ is rational & $\sqrt{q}$ is a surd) then the other root will be $p - \sqrt{q}$.

3. Common Roots of Two Quadratic Equations:

(a) At least one common root. Let $\alpha$ be the common root of $ax^2 + bx + c = 0$ & $a'x^2 + b'x + c' = 0$ then:
$a\alpha^2 + b\alpha + c = 0$ $a'\alpha^2 + b'\alpha + c' = 0$

By Cramer’s Rule: $\frac{\alpha^2}{bc'-b'c} = \frac{\alpha}{a'c-ac'} = \frac{1}{ab'-a'b}$

Therefore, $\alpha = \frac{ca'-c'a}{ab'-a'b} = \frac{bc'-b'c}{a'c-ac'}$

So the condition for a common root is: $(bc' - b'c)^2 = (ab' - a'b)(ac' - a'c)$

(b) If both roots are same then: $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$

4. Roots Under Particular Cases:

Let the quadratic equation $ax^2 + bx + c = 0$ has real roots and: (a) If $b = 0 \Rightarrow$ roots are of equal magnitude but of opposite sign.

(b) If $c = 0 \Rightarrow$ one root is zero other is $-\frac{b}{a}$.

(c) If $a = c \Rightarrow$ roots are reciprocal to each other.

(d) If $ac < 0 \Rightarrow$ roots are of opposite signs.

(e) If $a > 0, b > 0, c > 0$ or $a < 0, b < 0, c < 0 \Rightarrow$ both roots are negative.

(f) If $a > 0, b < 0, c > 0$ or $a < 0, b > 0, c < 0 \Rightarrow$ both roots are positive.

(g) If sign of $a =$ sign of $b \neq$ sign of $c \Rightarrow$ Greater root in magnitude is negative.

(h) If sign of $b =$ sign of $c \neq$ sign of $a \Rightarrow$ Greater root in magnitude is positive.

(i) If $a + b + c = 0 \Rightarrow$ one root is 1 and second root is $\frac{c}{a}$.

5. Maximum & Minimum Values of Quadratic Expression:

Maximum or Minimum Values of expression $y = ax^2 + bx + c$ is $\frac{-\Delta}{4a}$ which occurs at $x = -\frac{b}{2a}$ according as $a < 0$ or $a > 0$.
$y \in [\frac{-D}{4a}, \infty)$ if a > 0 & $y \in (-\infty, \frac{-D}{4a}]$ if a < 0

6. Location of Roots:

Let $f(x) = ax^2 + bx + c$, where $a, b, c \in \mathbb{R}, a \neq 0$: (a) Conditions for both the roots of $f(x) = 0$ to be greater than a specified number ‘d’ are $\Delta \geq 0$ and $a \cdot f(d) > 0$ & $-\frac{b}{2a} > d$.

(b) Condition 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 $a \cdot f(d) < 0$.

(c) Condition for exactly one root of $f(x) = 0$ to lie in the interval $(d,e)$ i.e. $d < x < e$ is $f(d) \cdot f(e) < 0$.

(d) Conditions that both roots of $f(x) = 0$ to be confined between the numbers $d$ & $e$ are: $\Delta \geq 0$ and $a \cdot f(d) > 0$ & $a \cdot f(e) > 0$ and $d < -\frac{b}{2a} < e$.

7. General Quadratic Expression in Two Variables:

$f(x, y) = ax^2 + 2hxy + by^2 + 2gx + 2fy + c$ may be resolved into two linear factors if:
$D = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ OR $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0$

8. Theory of Equations:

If $\alpha_1, \alpha_2, \alpha_3, \ldots, \alpha_n$ are the roots of the equation: $f(x) = a_0x^n + a_1x^{n-1} + a_2x^{n-2} + \ldots + a_{n-1}x + a_n = 0$ where $a_0, a_1, \ldots, a_n$ are constants $a_0 \neq 0$ then:

• $(i) \sum \alpha_1 = -\frac{a_1}{a_0}$
• $(ii) \sum (\alpha_1 \alpha_2) = +\frac{a_2}{a_0}$
• $(iii) \sum (\alpha_1 \alpha_2 \alpha_3) = -\frac{a_3}{a_0}$
• $(iv) \sum (\alpha_1 \alpha_2 \alpha_3....\alpha_n) = (-1)^n\frac{a_n}{a_0}$

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