Question

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

Quadratics
ANSWERED
asked 2021-03-08
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 \(\displaystyle{x}\in{R}\)

Answers (1)

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,
\(\displaystyle{x}=\frac{{1}}{{2}},{x}=-{3},{x}={1}.\)

0
 
Best answer

expert advice

Need a better answer?
...