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Nature of Roots, Relation Between Roots and Coefficient of Quadratic Equation is considered one of the most asked concept.
86 Questions around this concept.
If has real roots and , where then
The number of real solutions of the equation , is
Let the function have a maxima for some value of and a minima for some value of . Then, the set of all values of is
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Let be a real number. Let be the roots of the equation and be the roots of the equation . Then are the roots of the equation
The number of real roots of the equation , is
If for , then and are the roots of the equation:
Let $S$ be the set of positive integral values of a for which $\frac{a^2+2(a+1) x+9 a+4}{x^2-8 x+32}<0, \forall x \in \mathbb{R}$. Then, the number of elements in $\mathrm{S}$ is:
The sum of all the roots of the equation is:
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then:
Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{px}^2+\mathrm{qx}-\mathrm{r}=0$, where $\mathrm{p} \neq 0$. If $\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :
Let the quadratic equation is ax2 + bx + c = 0, (a,b,c ∈ R)
D (called the discriminant of the equation) = b2 - 4ac
Roots of this equation are given by
i) if D < 0, then both roots are non-real (imaginary numbers), and the roots will be conjugate of each other, means if p + iq is one of
the roots then other root will be p - iq
ii) If D > 0, then roots will be real and distinct
iii) if roots D = 0, then roots will be real and equal, and they equal
Special cases of case ii (D > 0)
i) if a,b,c are rational numbers (Q) and
If D is perfect square, then roots are rational
If D is not perfect square than roots are irrational (in this case if is one root of quadratic equation then other root will be )
ii) If a = 1 and b and c are integers and
If D is perfect square, then roots are integers
If D is not perfect square than roots are non-integrer values
Let ? and ? be two roots of a quadratic equation. So, we have
Sum of roots:
Product of roots:
The difference of root can also be found in the same way by manipulating the terms
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