Question

Use the factorization method and solve the following equation \[4{x^2} - 4x = 15\].

Answer 1

Hint: In order to solve the given expression by factorization method we will multiply the \[{x^2}\]term with the constant term and express the resultant in two numbers such that their addition or subtraction is equal to the middle term.

Complete step-by-step solution:
The following are the steps involved in using the factorization method
Given equation,
\[4{x^2} - 4x = 15\]
Or
 \[4{x^2} - 4x - 15 = 0 - - - - - (1)\]
Firstly, we multiply the coefficient of \[x\]that is \[4\] with the constant term that is\[ - 15\], we get\[ - 60{x^2}\]
Now, we express \[{x^2}\]as a product of two number whose sum or difference is equal to the middle term
Thus, the value \[ - 60{x^2}\]can be written as
\[( - 10x)(6x)\]
And we know that \[( - 10x) + (6x) = - 4x\]
Now, we rewrite the equation in such a manner that common factors can be taken out from the first two terms and the last two terms.
We get,
\[4{x^2} - 10x + 6x - 15 = 0 - - - - - (2)\].
Taking out the common factors we get,
\[2x(2x - 5) + 3(2x - 5) = 0 - - - - - (3)\]
Further,
\[(2x - 5)\left( {2x + 3} \right) = 0 - - - - - (4)\]
Thus, either \[(2x - 5) = 0\] or \[\left( {2x + 3} \right) = 0\]

Hence, \[x = \dfrac{5}{2} \]or \[\dfrac{{ - 3}}{2}\]

Additional information: There is no shortcut to using the factorization method however, with little practice one can easily master it. There is another systematic way of solving a quadratic equation, it is by using the quadratic formula, which is,\[\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. In which for an equation of the form \[a{x^2} + bx + c = 0\]a, b, and c are the coefficients.

Note: Factorising an equation is an exact opposite of expanding the brackets. Therefore, to check if you have the questions correctly then you should expand the equation \[(4)\]and see if you get the exact equation as in\[(1)\].
Let’s check,
We have equation (4) as
\[(2x - 5)\left( {2x + 3} \right) = 0\]
Multiplying both of the above brackets,
\[ \Rightarrow 2x \times 2x + 2x \times 3 - 5 \times 2x - 15 = 0\]
\[ \Rightarrow 4{x^2} + 6x - 10x - 15 = 0\]
\[ \Rightarrow 4{x^2} - 4x - 15 = 0 \Rightarrow 4{x^2} - 4x = 15\] which is our required equation. hence, we have solved the question correctly.