Find the roots of the function f(x)=(2x−1)⋅(x2+2x−3), with x∈R

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High School
Algebra
Algebra I
Quadratic function and equation

rocedwrp

Answered question

2021-03-08

Find the roots of the function

f (x) = (2 x - 1) \cdot (x^{2} + 2 x - 3)

, with

x \in R

Answer & Explanation

StrycharzT

Skilled2021-03-09Added 102 answers

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

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