Question

How do you solve the equation $4{x^2} - 12x = 7$ algebraically for $x$ ?

Answer 1

Hint: First, identify the middle term of the given expression. Then, break the middle term into such numbers so that its product is equal to the product of the first and third term and those two numbers when added give the original middle term. Once we have done that, simply do grouping and take out the common factors and get our required factors for the sum. Then we equate each factor with zero to find the value of $x$.

Complete step-by-step solution:
The given expression is $4{x^2} - 12x = 7$, we subtract both sides with $ - 7$ to bring it in the standard quadratic form which is given as: $4{x^2} - 12x - 7 = 0$. This is now a trinomial, which means an expression with three terms. We need to factorize this given expression to get our required values of $x$.
Factoring an expression simply means to find the roots of $x$ for which the equation gives the value as zero on solving. In order to do so, we first identify our middle term.
The middle term in the given expression is: $ - 12x$ , now we need to break the middle term into two such numbers so that its equal to the product of the first and third term of the given expression and adds up to give the original middle term.
First term of the expression: $4{x^2}$
Third term of the expression: $ - 7$
Product of the first and third term: $ - 28{x^2}$
In order to break the middle term, into two numbers – we take the L.C.M of the product of the first and third term to get our required numbers:
$\begin{gathered}
  {\text{ }}2\left| \!{\underline {\,
  {28} \,}} \right. \\
  {\text{ }}2\left| \!{\underline {\,
  {14} \,}} \right. \\
  7\left| \!{\underline {\,
  7 \,}} \right. \\
  \underline 1 \\
\end{gathered} $
The factors of $28$ are: $2 \times 2 \times 7$
Now, we need to find two such numbers which when multiplied together gives $28$. Thus, we take the two numbers as: $\left( {2 \times 7} \right) \times 2 = 14 \times 2$
Now, if we add $ - 14$ and$ + 2$ , we get $ - 12$
Thus, we have our required two numbers, we just add the variable $x$ to them.
Thus, our new expression becomes: $4{x^2} - 14x + 2x - 7$
Now we do further grouping by taking out the common factors:
$\Rightarrow$$2x\left( {2x - 7} \right) + 1\left( {2x - 7} \right)$
We see that $\left( {2x - 7} \right)$ is a common factor in the above expression, hence it’s our first factor while the other factor is the common terms:
Thus, the two factors are: $\left( {2x - 7} \right)\left( {2x + 1} \right)$
Now, let us equate each factor with zero to get our required value of $x$ .
For the first factor:
$\Rightarrow$$2x - 7 = 0$
Adding $7$ to both sides, we get:
$\Rightarrow$$2x = 7$
On simplifying further:
$\Rightarrow$$x = \dfrac{7}{2}$
For the second factor:
$\Rightarrow$$2x + 1 = 0$
Adding $ - 1$ to both sides, we get:
$\Rightarrow$$2x = - 1$
On simplifying further:
$\Rightarrow$$x = - \dfrac{1}{2}$

Thus, the values of $x$ are $\dfrac{7}{2}$ and $ - \dfrac{1}{2}$.

Note: Factorization is simply the method of breaking down a given expression into their simplest factors. Grouping method used to solve the sum above is the most common method of solving any quadratic equation. One should also be attentive towards the signs placed in front of the numbers, as many students make a mistake in getting confused with the signs. You can also solve the given equation by –
Using Quadratic Formula
Completing the square method