**Problem 1:**

Find the condition that a quadratic function of x and y may be resolved into two linear factors. For instance, a general form of such a function would be : .

**Problem 2:**

Find the condition that the equations and may have a common root.

Using the above result, find the condition that the two quadratic functions and may have a common linear factor.

**Problem 3:**

For what values of m will the expression be capable of resolution into two rational factors?

**Problem 4:**

Find the values of m which will make equivalent to the product of two linear factors.

**Problem 5:**

Show that the expression always admits of two real linear factors.

**Problem 6:**

If the equations and have a common root, show that it must be equal to or .

**Problem 7:**

Find the condition that the expression and may have a common linear factor.

**Problem 8:**

If the expression can be resolved into linear factors, prove that P must be be one of the roots of the equation .

**Problem 9:**

Find the condition that the expressions and may be respectively divisible by factors of the form and .

**Problem 10:**

Prove that the equation for every real value of x, there is a real value of y, and for every real value of y, there is a real value of x.

**Problem 11:**

If x and y are two real quantities connected by the equation , then will x lie between 3 and 6, and y between 1 and 10.

**Problem 11:**

If , find the condition that x may be a rational function of y.

More later,

Regards,

Nalin Pithwa.

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