Math topic中文

Always positive negative

conditions for $y = ax^{2} + bx + c$ to be always positive (or always negative)

Practice questions

1A curve has the equation $y = \frac{2x - 3}{3x + 4}$ , where $x \ne -\frac{4}{3}$ . The normal to the point $P$ on the cur…
2如果抛物线 $y=(k-4)x^{2}+k$ 的图都在 $x$ 轴上方，那么 $k$ 的取值范围是 ___.
3Solve the equation $2^{x+1}+2^{-x}=3$ . The equation of a curve is $y=2x^2-4ax+b$ where $a$ and $b$ are non-zero constant…
4A curve has the equation $y = x^3 + kx^2 + kx + 8$ . Find the set of values of $k$ such that the curve is an increasing f…
5若函数 $y=x^{3}+x^{2}+mx+1$ 是 $R$ 上的单调函数，则实数 $m$ 的取值范围是 $()$ A. $( \dfrac{1}{3} ,+∞)$ B. $(-∞, \dfrac{1}{3} ]$ C. $[ \dfrac{1}{3} ,+∞)$ …
6A curve has equation $y = x^2 + x + 2k - 4$ and a line has equation $y - 2x = k$ , where $k$ is a constant. Find the set…
7A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given…
8设函数 $f(x)=\left\{\begin{array}{l}2^{x},x\leqslant 0,\\log_{2}x,x>0,\end{array}\right.$ 对任意给定的 $y\in(2,+∞)$ ，都存在唯一的 $x\in R$ …
9The equation of a curve is $y = -2x^2 - kx - 6 + k$ , where $k$ is a constant. Find the range of values of $k$ for which…
10Given that $f(x)=x^2-ax+3$ , where $a$ is a constant, find the range of values of $a$ for which $f(x)>x-1$ , for all real…
11Find the range of values of $k$ such that $3x^2 - 5x + k$ is always positive. Hence, solve the inequality $\frac{x^2 - x - 2}{3x^2 - 5x + 4} < 0$ …
12Express $-9 + 6x - 2x^2$ in the form $a(x+b)^2 + c$ and hence explain why $-9 + 6x - 2x^2$ is negative for all real valu…
13Express $y = 4x - 4x^2 - 3$ in the form $p(x + q)^2 + r$ where $p$ , $q$ and $r$ are constants. Hence, explain whether $4x - 4x^2 - 3 = 0$ …
14Given that $4x^2 - \frac{x}{h} - 3 = 0$ , where $h \ne 0$ , show that there are no real values of $h$ for which the equati…
15Answer all questions. Find the set of values of the constant $k$ for which $y = kx - 2$ meets the curve $y^2 = 4x - x^2$ …
16The line of symmetry of a quadratic curve is $x = -2$ and the curve lies above the $x$ -axis for all $x$ . Given that the…
17已知关于 $x$ 的不等式 $ax^{2}+4ax-3< 0.(1)$ 若不等式的解集为 $\left\{\begin{array}{l}x\left|{x< -3}\right.\end{array}\right.$ 或 $x>-1\}$ …
18Given that the curve $y = ax^2 + 2bx - 1$ lies entirely below the x-axis, determine the conditions that must be applied…
19Find the range of values of $k$ for which $x^2 + 12x + 9$ is always greater than $4x + k$ .
20Find the range of values of $x$ which satisfies $\frac{x^2 + 2x + 2}{x^2 - 5x + 6} \ge 0.$ Show that the line $y = x + k$ …
21A curve has the equation $y = x^2 + px - 4p + 9$ , where $p$ is a constant. Find the range of values of $p$ such that $x^2 + px - 4p + 9$ …
22Express $y = 2x^2 - 4x + 7$ in the form $a(x - b)^2 + c$ , where $a$ , $b$ and $c$ are constants. Hence, explain why the c…
23Find the range of values of the constant $p$ such that $y = px^2 - 4x + p - 3$ is always positive for all real values of…
24Find the range of values of $m$ for which $x^2 - 2mx + 2m$ is greater than $-3$ for all real values of $x$ .
25Find the range of values of $k$ for which $kx^2 + 8x + k < 6$ for all real values of $x$ . Given the curve $y = 8x^2 + mx + m - 6$ …
26Find the range of values of $k$ for which the line $2x - y = 5$ intersects the curve $xy = kx - 2$ at two distinct point…
27Find the range of values of the constant $m$ for which the curve $y = (m - 3)x^2 + mx + m$ lies completely above the $x$ …
28Find the values of $k$ for which the curve $y = 2x^2 + (3k + 1)x + 4$ lies entirely above the line $y = 2x - 3k^2 + 5$ .
29The equation of a curve is $y = (m+1)x^2 - 8x + 3m$ , where $m$ is a constant. In the case where $m = 1$ , show that the l…
30Explain by completing the square that $x^2 - 2x + 3$ is always positive for all real values of $x$ . Hence, find the rang…
31Find the range of values of the constant $k$ for which $k\left(x^2 + x\right) + 2\left(x^2 + x + 1\right) = 0$ has no re…