Ormezzani6cuu

2022-04-07

Finding a Quadratic Function Find a quadratic function f (with integer coefficients) that has the givenzeros. Assume that b is a positive integer and a is an integer not equal to zero.
(a) $±\sqrt[\left(\right)]{b}i$
(b) $a±bi$

slaastro132z

Step 1
To determine:
Find a Quadratic function f (with integer coefficients) that has the given zeroes. Assume that b is a positive integer and a is an integer not equal to zero.
(a) $±\sqrt[\left(\right)]{b}i$
(b) $a±bi$
Step 2: Calculation
Part a)
Consider $x=±\sqrt[\left(\right)]{b}i$ be the zeroes of the quadratic function.
Since, $x=\sqrt[\left(\right)]{b}i$ is the zero of the quadratic function, therefore, $\left(x-\sqrt[\left(\right)]{b}i\right)$ is a factor of quadratic function.
Also, since, $x-\sqrt[\left(\right)]{b}i$ is the zero of the quadratic function, therefore, $\left(x+\sqrt[\left(\right)]{b}i\right)$ is a factor of quadratic function.
So, the quadratic function is given by:
$\left(x-\sqrt[\left(\right)]{b}i\right)\left(x+\sqrt[\left(\right)]{b}i\right)$
$=\left({x}^{2}-{\left(\sqrt[\left(\right)]{b}i\right)}^{2}\right)\left[\because \left(a-b\right)\left(a+b\right)\right]$
$={x}^{2}-b{\left(i\right)}^{2}$
$={x}^{2}-b\left(-1\right)\left[\because {i}^{2}=-1\right]$
$={x}^{2}+b$

Part b)
Consider $x=a±bi$ be the zeroes of the quadratic function.
Since, $x=a+bi$ is the zero of the quadratic function, therefore, $\left(x-\left(a+bi\right)\right)$ is a factor of quadratic function.
Since, $x=a-bi$ is the zero of the quadratic function, therefore, $\left(x-\left(a-bi\right)\right)$ is a factor of quadratic function.
So, the quadratic function is given by:
$\left(x-\left(a+bi\right)\right)\left(x-\left(a-bi\right)\right)$
$=\left(\left(x-a\right)-bi\right)\left(\left(x-a\right)+bi\right)$
$=\left({\left(x-a\right)}^{2}-{\left(bi\right)}^{2}\right)\left[\because \left(a-b\right)\left(a+b\right)\right]$
$={x}^{2}+{a}^{2}-2ax-{b}^{2}{\left(i\right)}^{2}$
$={x}^{2}+{a}^{2}-2ax-{b}^{2}\left(-1\right)\left[\because {i}^{2}=-1\right]$
$={x}^{2}-2ax+{a}^{2}+{b}^{2}$

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