Quadratic forms and equations

Example

\(x^2-6x+5=(x-1)(x-5)=(x-3)^2-4\). Zeros are \(1,5\), the vertex is \((3,-4)\), and the minimum value is \(-4\).

Solve by factorising, completing the square or \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). For \(x^2-6x+5=4\), completing the square gives \((x-3)^2=8\), hence \(x=3\pm2\sqrt2\).

Choosing the useful quadratic form

Use factorised form when the roots are known, vertex form when a turning point is known, and expanded form when the vertical intercept is given. A point away from the roots or vertex determines the remaining scale factor \(a\). Check the sign of \(a\) against the opening direction.

The axis of symmetry is halfway between two real roots: \(x=\dfrac{p+q}{2}\). If one root and the axis are known, reflect that root across the axis to obtain the other. Substitute the axis value into the function to find the vertex height.

For a model, first state the permitted inputs and their units. A vertex outside this interval cannot supply the model’s maximum or minimum; compare the permitted endpoint values instead. A negative root may be mathematically valid but outside a time or length model.

For \(a\ne0\), completing the square gives \(ax^2+bx+c=a\left(x+\dfrac{b}{2a}\right)^2+c-\dfrac{b^2}{4a}\). On the full real domain, the range is \([c-\dfrac{b^2}{4a},\infty)\) when \(a>0\), and \((-\infty,c-\dfrac{b^2}{4a}]\) when \(a<0\). The vertex is a minimum in the first case and a maximum in the second.

An expression has a value; an equation asks which inputs make a statement true. To use the zero-product principle, first make one side zero: \(AB=0\) implies \(A=0\) or \(B=0\). Factorising rather than dividing by an expression containing \(x\) keeps possible zero-factor solutions. State all roots and check them in the original equation.

Use an obvious factor pair or common factor first; completing the square is useful for a monic quadratic or a requested vertex form; the quadratic formula works when factorisation is inconvenient. In the formula, include the signs of \(a,b,c\), bracket negative coefficients, and divide the whole numerator by \(2a\).

Keep square roots and calculator values unrounded until the final answer. Use \(=\) for exact steps and \(\approx\) for approximations. A terminating decimal may be exact; a rounded root should satisfy the original equation approximately. Check algebraic restrictions using an exact expression. Follow the stated accuracy; if none is specified, prefer exact values when convenient and otherwise give 3 significant figures.