The quadratic formula

Identifying the coefficients

Write the equation as \(ax^2+bx+c=0\), where \(a\ne0\). Include the signs when identifying \(a\), \(b\) and \(c\).

\(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)

The denominator \(2a\) divides the whole numerator. Use brackets around negative coefficients when substituting.

Accuracy and checking

Evaluate each branch using the unrounded square-root value. In working, use \(=\) between exact expressions and \(\approx\) for approximate calculations. A final answer written as \(x=2.63\) (2 d.p.) is understood to be rounded to the stated accuracy.

A result such as \(x=1.25\) is exact if it is a terminating root; a rounded root should satisfy the original equation approximately. Keep an exact expression when checking algebraic restrictions.

For this chapter, follow each question’s stated accuracy. Where none is specified, give exact answers when convenient; otherwise use 3 significant figures.