Question

How do I use the quadratic formula to find the zeros of $f$ ?

Answer 1

Step 1: Find zeroes of a polynomial using the Quadratic formula.

Given: A function of the form $f\left(x\right)=a{x}^2+bx+c$, one can find the zeroes function (that is, where $f\left(x\right)=0$) by using the quadratic formula $=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

If the discriminant $b^2-4ac$ is under nothing, these roots will be perplexing or nonexistent, and if the discriminant is more prominent than or equivalent to $0$. these roots will be genuine.

The use of $\pm$ illuminates us that there are two arrangements here; one where we are taking away $\sqrt{b^2-4ac}$ and one where we are adding it.

Step 2: Assume an example for better understanding.

For a bit-by-bit model, expect the capability $f\left(x\right)=x^2-7x+10$. Here $a=1,b=-7,c=10$.

The values of $x$ for which $f\left(x\right)=0$, in other words, the root of $f\left(x\right)$, would be decided by the quadratic equation:

$\begin{array}{rcl}x& =& \frac{-\left(-7\right)\pm \sqrt{{\left(-7\right)}^2-4\left(1\right)\left(10\right)}}{2\left(1\right)}\\ & =& \frac{7\pm \sqrt{49-40}}{2}\\ & =& \frac{7\pm \sqrt{9}}{2}\\ & =& \frac{7\pm 3}{2}\end{array}$.

Thus, the roots will be at $x=\frac{7+3}{2}$ and $x=\frac{7-3}{2}$ or $x=5$ and $x=2$.

Hence, on the graph of the function, we would see that the parabola crosses the x-axis at $x=2$ and $x=5$.