Question

How to convert quadratic equation from standard form to vertex form ?

Answer 1

Convert quadratic equation from standard form to vertex form

A quadratic equation is given as, $y=a{x}^2+bx+c\left\{a,b,c\in \mathrm{ℝ},\mathrm{a}\ne 0\right\}$.

From the above equation, take $a$ as common from the terms containing $x$.

$\Rightarrow y=a\left(x^2+\frac{bx}{a}\right)+c$

Complete the square inside the bracket by adding and subtracting ${\left(\frac{b}{2a}\right)}^2$

$$\begin{gathered} \Rightarrow y=a\left[x^2+\frac{bx}{a}+{\left(\frac{b}{2a}\right)}^2-{\left(\frac{b}{2a}\right)}^2\right]+c \\ \Rightarrow y=a\left[{\left(x+\frac{b}{2a}\right)}^2-\frac{b^2}{4a^2}\right]+c \\ \Rightarrow y=a{\left(x+\frac{b}{2a}\right)}^2-\frac{b^2}{4a}+c \\ \Rightarrow y=a{\left(x-h\right)}^2+k \end{gathered}$$

where, $h=-\frac{b}{2a}$ and $k=c-\frac{b^2}{4a}$

Hence, the vertex form of a quadratic equation is $\mathrm{y}=a{\left(x-h\right)}^2+\mathrm{k}$where, $h=-\frac{b}{2a}$ and $k=c-\frac{b^2}{4a}$.