Question

# Find the roots of the function f(x) = (2x − 1)*(x^2 + 2x − 3), with x in R

Find the roots of the function
$$\displaystyle{f{{\left({x}\right)}}}={\left({2}{x}−{1}\right)}\cdot{\left({x}^{{2}}+{2}{x}−{3}\right)}$$, with $$\displaystyle{x}\in{R}$$

2021-03-09

First find the factors of $$\displaystyle{\left({x}^{{2}}+{2}{x}-{3}\right)}.$$
Compare the $$\displaystyle{a}{x}^{{2}}+{b}{x}+{c}$$ and $$\displaystyle{x}^{{2}}+{2}{x}-{3}$$, implies that $$a = 1, b = 2$$, and $$c = -3$$.
The ac-method of factoring quadratics: To find factors of quadratic $$\displaystyle{a}{x}^{{2}}+{b}{x}+{c}$$, Find two numbers whose sum is b and product is $$a\cdot c$$.
Here $$a\cdot c = 1\cdot (-3) = -3$$and $$b = 2$$.
To find factors of quadratic $$\displaystyle{x}^{{2}}+{2}{x}-{3}$$, find two numbers whose sum is 2 and the product is -3.
Such numbers are 3 and -1.
The factors of $$\displaystyle{\left({x}^{{2}}+{2}{x}-{3}\right)}$$ are $$(x + 3)(x - 1).$$
The given function $$\displaystyle{f{{\left({x}\right)}}}={\left({2}{x}−{1}\right)}\cdot{\left({x}^{{2}}+{2}{x}−{3}\right)}$$ becomes
$$\displaystyle{f{{\left({x}\right)}}}={\left({2}{x}−{1}\right)}·{\left({x}+{3}\right)}·{\left({x}-{1}\right)}$$
Set $$f(x) = 0$$, implies that
$$\displaystyle{\left({2}{x}−{1}\right)}\cdot{\left({x}+{3}\right)}\cdot{\left({x}-{1}\right)}={0}$$
By using zero product property,
$$\displaystyle{\left({2}{x}−{1}\right)}={0},{\left({x}+{3}\right)}={0},{\left({x}-{1}\right)}={0}$$
Implies that,
$$2x = 1, x = -3, x = 1$$
That is, the roots of the given function are,
$$\displaystyle{x}=\frac{{1}}{{2}},{x}=-{3},{x}={1}.$$