Sometimes the equation contains an unknown coefficient, and instead you are told a fact about its roots — the sum of their squares, that they differ by a fixed amount, or that they are reciprocals of each other. Translate the fact into an equation in \(\alpha+\beta\) and \(\alpha\beta\), then solve for the unknown.**Translating conditions**Solving for the unknown may give more than one value. Check every value against any extra information — for example that the roots are real, or that both roots are positive.If both roots are positive, then both \(\alpha+\beta>0\) and \(\alpha\beta>0\). Use these signs to reject invalid values of the unknown.

Sometimes the equation contains an unknown coefficient, and instead you are told a fact about its roots — the sum of their squares, that they differ by a fixed amount, or that they are reciprocals of each other. Translate the fact into an equation in \(\alpha+\beta\) and \(\alpha\beta\), then solve for the unknown.

Working backwards

Solving for the unknown may give more than one value. Check every value against any extra information — for example that the roots are real, or that both roots are positive.

If both roots are positive, then both \(\alpha+\beta>0\) and \(\alpha\beta>0\). Use these signs to reject invalid values of the unknown.

Condition on the roots	Equation to use
Sum of squares equals \(k\)	\((\alpha+\beta)^2-2\alpha\beta=k\)
Roots differ by \(d\)	\((\alpha+\beta)^2-4\alpha\beta=d^2\)
Roots are reciprocals of each other	\(\alpha\beta=1\)
One root is double the other	\(\beta=2\alpha\): sum \(=3\alpha\), product \(=2\alpha^2\)
Roots are equal in magnitude but opposite in sign	\(\alpha+\beta=0\)

Finding Unknown Coefficients from Root Conditions

Finding Unknown Coefficients from Root Conditions