Question

How do you write \[3{x^2} - x - 4\] in factored form?

Answer 1

Hint: The standard form of a quadratic equation is $a{x^2} + bx + c = 0$ . To find the factors of the given equation, we compare the given equation and the standard equation and get the values of a, b and c. Then we will try to write b as a sum of two numbers such that their product is equal to the product of a and c, that is, ${b_1} \times {b_2} = a \times c$ , this method is known as factorization. We find the value of ${b_1}$ and ${b_2}$ by hit and trial. We move to some other methods like quadratic formula, graphing, and completing the square method if we are not able to solve an equation by factorization.

Complete step-by-step solution:
We have to factorize \[3{x^2} - x - 4\] , so we factorize it as follows –
\[
  3{x^2} - x - 4 \\
   \Rightarrow 3{x^2} + 3x - 4x - 4 \\
   \Rightarrow 3x(x + 1) - 4(x + 1) \\
   \Rightarrow (3x - 4)(x + 1) \\
 \]

Hence the factored form of the equation \[3{x^2} - x - 4\] is $(3x - 4)(x + 1)$ .

Note: In this question, a polynomial equation of degree 2 is given to us as 2 is the highest exponent in the given equation, hence it is a quadratic equation and has exactly two solutions. Solutions of an equation are defined as the values of the x for which the given function has a value zero or we can say that the solutions of a function are the points on which the y-coordinate is zero when we plot the function on the graph, thus they are simply the x-intercepts. But in this question, we are asked to only factorize it. Its solutions can be obtained by putting the obtained equation equal to 0 –
\[
  3x - 4 = 0,\,x + 1 = 0 \\
   \Rightarrow x = \dfrac{4}{3},\,x = - 1 \\
 \]