Standard form of Quadratic Equation

Question

$Ax^2 + Bx + C = 0$ is considered the standard form of a quadratic equation. What exactly is the form of an equation? Is the form of the following two quadratic equations the same: 2x^2 - 6 = 0, 2a^2 - 6 = 0. If yes then please explain how. Thanks.

Answer 1

A 'form' of an equation is a way to write an expression. The standard form of a quadratic is $y=ax^2+bx+c$ because it expresses the coefficients of the powers of $x$. This makes it easy to apply the quadratic formula.

Different forms of quadratics can be useful for different things. For example, the same quadratic can be expressed as $y=a(x-h)^2+k$, where $(h,k)$ is the turning point. Root form is the factorised version of standard form, which is given by $y=(x-a)(x-b)$. Root form clearly indicates the $x$-intercepts of the quadratic. Note that you can rearrange the equation to achieve any of these forms.

This isn't limited to quadratics. Think about cubics, square root functions, inverse relations, and even trigonometric equations have their own standard forms.

Answer 2

The standard form of an equation is how it is most commonly expressed, since it is usually easier to deal with and to see properties of it.

For the quadratic equation, the usual/standard form is $$ax^2+bx+c=0$$ since we can easily see the coefficients of each power of $x$ and plug them into the quadratic formula. Writing it like $$x^2+\frac ba x+\frac ca=0\quad\text{or}\quad x(ax+b)=-c$$ makes it harder to do this.

This applies to any equation/formula.

For example, the area of a circle is given by $$A=\pi r^2$$ not $$A=\pi\left(\frac d2\right)^2=\frac14\pi d^2$$ since this is more complicated and rather unnecessary.