Solve for $𝑥$. If there are multiple answers, separate them with commas. Give answers as fractions and not decimals.
$\frac{3}{10000}x^2-\frac{7}{100}x+4=0$
$\left(x+3\right)^2=\frac{x+3}{2}+3$
$x^2-\left(\sqrt{2}+\sqrt{3}\right)x+\sqrt{6}=0$
$x^2-\left(a-2\right)x-2a=0$ (where $a$ is a constant)
$ax^2-\left(a+b\right)x+b=0$ (where $a$ and $b$ are constants)
$\left(a-1\right)x^2-\left(a^2-1\right)x=0$ (where $a$ is a constant)
The quadratic equation $x^2+2ax-3a=0$ has 2 solutions. One of the solutions is $x=-3$. Find $a$ and the other value of $x$.
$a=$
$x=$
If the quadratic equation $x^2-ax+24=0$ has 2 integer solutions, what is the minimum value of $a$? Assume $a$ is positive.
$a=$
The quadratic equation $x^2+2ax+a^2-4=0$ has 2 solutions. The difference between the two solutions is $4$ and one solution is $5$ times greater than the other solution. Find the value of $a$ and the two values of $x$ (separate them with a comma).
$a=$
$x=$
The graph of the function $f\left(x\right)=2x^2+px+q$ passes through the point $\left(1,3\right)$ and its vertex lies on the line $y=2x-3$.
Find the two sets of values of $p$ and $q$.
$p=$
$, q=$
or $p=$
$, q=$
The graphs of the functions $f\left(x\right)=-2x^2+\left(a+6\right)x-b$ and $g\left(x\right)=3x^2-3\left(b-1\right)x+7a+1$ share the same vertex.
Find the two sets of values of $a$ and $b$.
$a=$
$, b=$
or $a=$
$, b=$
A piece of wire of length $22$cm is cut into two pieces. One piece is bent in to an equilateral triangle and the other piece is bent into a square. Let $A$ be the sum of the areas of triangle and square. Find the length of the sides of the square that gives the minimum value of $A$.
Give your answer to $3$ decimal places:
cm
If $x^2-y+2=0$, find the minimum value of $x^2+y^2$.
Point $C$ is on line segment $AB$ so that a square with sides of length $AC$ and an equilateral triangle with sides of length $BC$ is a minimum. Find the value of $\frac{AC}{BC}$ in simplest form.
Find the integer values of $a$ such that $\displaystyle\frac{x^2+x+a−5}{x−1}=a$ has no real root.
*Separate your answers with commas
Equilateral triangle $\triangle ABC$ has sides of length $a$. Points $P,Q$ and $R$ lie on sides $BC,CA$ and $AB$ such that $BP:CQ:AR=1:2:3$.
Find the minimum area of $\triangle PQR$ in terms of $a$.