Recent questions tagged quadratic-equations

1 votes
1 answer
1
1 votes
1 answer
3
Let $\text{m}$ and $\text{n}$ be positive integers, If $x^{2} + mx + 2n = 0$ and $x^{2} + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is $7$...
0 votes
0 answers
4
In how many ways can a pair of integers $\textsf{(x , a)}$ be chosen such that $x^{2} – 2 |x| + |a-2| = 0 ?$$4$ $5$$6$$7$
1 votes
1 answer
5
0 votes
1 answer
6
The expression $(11.98\times 11.98 + 11.98 \times x +0.02 \times 0.02)$ will be a perfect square for $x$ equal to:$2.02$$0.17$$0.04$$1.4$
0 votes
1 answer
7
1 votes
1 answer
10
The quadratic equation $x^{2}+bx+c=0$ has two roots $4a$ and $3a$, where a is an integer. Which of the following is a possible value of $b^{2}+c$?$3721$$549$$427$$361$
1 votes
1 answer
11
Let $A$ be a real number. Then the roots of the equation $x^{2}-4x-\log _{2}A=0$ are real and distinct if and only if$A \frac{1}{16}$$A \frac{1}{8}$$A< \frac{1}{16}$$A< \...
2 votes
1 answer
12
If $a$ and $b$ are integers such that $2x^{2}- ax + 2 0$ and $x^{2}-bx+8 \geq 0$ for all real numbers $x$, then the largest possible value of $2a-6b$ is _________
1 votes
1 answer
13
The minimum possible value of the sum of the squares of the roots of the equation $x^{2}+\left ( a+3 \right )x-\left ( a+5 \right )=0$ is$1$$2$$3$$4$
1 votes
1 answer
14
1 votes
1 answer
15
Given the quadratic equation $x^2 – (\text{A} – 3)x – (\text{A} – 2)$, for what value of $\text{A}$ will the sum of the squares of the roots be zero?$-2$$3$$6$$\t...
0 votes
0 answers
16
If both $a$ and $b$ belong to the set $\{1,2,3,4\}$, then the number of equations of the form $ax^2+bx+1=0$ having real roots is _______
0 votes
0 answers
17
If $ax^{2}+bx+c= 0$ and $2a,b$ and $2c$ are in arithmetic progression, which of the following are the roots of the equation?$a,c \\$$-a,-c \\$$-\dfrac{a}{2},-\dfrac{c}{2}...
1 votes
1 answer
19
1 votes
1 answer
20
For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive?$x^{2}-y^{2}=0$$(x-k)^{2}+y^{2}=1$$2$$0$$\s...
5 votes
3 answers
21
If one root of $x^{2} + px + 12 = 0$ is $4$, while the equation $x^{2} + px + q = 0$ has equal roots, then the value of $q$ is:$49/4$$4/49$$4$$\frac{1}{4}$
1 votes
1 answer
22
One root of $x^{2} + kx – 8 = 0$ is square of the other. Then, the value of k is:$2$ $8$ $-8$ $-2$
–1 votes
1 answer
23
Given the quadratic equation $x^{2}-(A-3) x- (A-2) = 0$, for what value of $A$ will the sum of the squares of the roots be zero?$-2$$3$ $6$ None of these
0 votes
1 answer
24
If both $a$ and $b$ belong to the set $\{1, 2, 3, 4\}$, then the number of equations of the form $ax^2 + bx + 1 = 0$ having real roots is$10$$7$$6$$12$
0 votes
1 answer
27
Let $f(x) = ax^2 + bx +c$, where $a, b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2)$ and that $3$ is a root of $f(x)=0$.What is the val...
0 votes
0 answers
29
The number of roots of $\frac{A^2}{x} + \frac{b^2}{x-1} =1$ is $1$$2$$3$None of these
0 votes
0 answers
30
Let $p$ and $q$ be the roots of the quadratic equation $x^2 - (a-2) x-a -1 =0.$ What is the minimum possible value of $p^2 + q^2?$$0$$3$$4$$5$
0 votes
0 answers
31
Let $f(x) = ax^2 - b |x|$, where $a$ and $b$ are constants. Then at $x=0, f(x)$ is, maximized whenever $a>0, b>0$maximized whenever $a>0, b<0$minimized whenever $a>0, b>0...
0 votes
1 answer
32
For which value of $k$ does the following pair of equations yield a unique solution for $x$ such that the solution is positive?$x^2 - y^2 =0$$(x-k)^2 + y^2 =1$$2$$0$$\sqr...
2 votes
1 answer
33
A quadratic function f(x) attains a maximum of $3$ at $x=1.$ The value of the function at $x=0$ is $1.$ What is the value of $f(x)$ at $x=10?$$-119$$-159$$-110$$-180$$-10...
2 votes
1 answer
34
Let $f(x) = ax^2 + bx +c$, where $a$, $b$ and $c$ are certain constants and $a \neq 0$. It is known that $f(5) = -3 f(2) $ and that 3 is a root of $f(x)=0$.What is the ot...
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