Basic Technical Mathematics, 11th Edition

14.3 Equations in Quadratic Form

• Substituting to Fit Quadratic Form • Solving Equations in Quadratic Form • Extraneous Roots

Often, we encounter equations that can be solved by methods applicable to quadratic equations, even though these equations are not actually quadratic.

NOTE

[They do have the property, however, that with a proper substitution they may be written in the form of a quadratic equation.]

All that is necessary is that the equation have terms including some variable quantity, its square, and perhaps a constant term. The following example illustrates these types of equations.

EXAMPLE 1 Identifying quadratic form

1. The equation $x\hspace{0.17em}-\hspace{0.17em}2\sqrt{x}\hspace{0.17em}-\hspace{0.17em}5\hspace{0.17em}=\hspace{0.17em}0$ is an equation in quadratic form, because if we let $y\hspace{0.17em}=\hspace{0.17em}\sqrt{x}\hspace{0.17em},\hspace{0.17em}$ we have $x\hspace{0.17em}=\hspace{0.17em}{(\sqrt{x})}^2\hspace{0.17em}=\hspace{0.17em}{y}^2$