Question

How do you solve ${x^2} - 3x - 28 = 0$?

Answer 1

Hint:This is a quadratic equation and it can be solved with a middle term factor. A middle term factor looks like $a{x^2} + bx + c$, where a, b and c are variables. We need to find the middle term in such a way that bx can be arranged and we can form a common pair, the first two terms, and the last two terms. Then we will continue simplifying.

Complete step by step solution:
The given equation is ${x^2} - 3x - 28 = 0$
Here we will 28 in such a way that after addition or subtraction we will get 3x.
$ \Rightarrow {x^2} - 7x + 4x - 28 = 0$
You can see that $ - 3x$ is written as $ - 7x + 4x$.
$ \Rightarrow x(x - 7) + 4(x - 7) = 0$
We simply took out the common terms
$ \Rightarrow (x - 7)(x + 4) = 0$
And now we have two different pairs, after multiplication, we will get the same equation as before i.e.,${x^2} - 3x - 28 = 0$.
As you can see the multiplication of these pairs is 0. So, any one of them must be equal to 0.
By equating $(x - 7) = 0$, we will get$x = 7$
And, by equating $(x + 4) = 0$, we will get $x = - 4$

So, the value of x can be 7 or -4.

Additional Information:
There are 3 forms of a quadratic equation, the above question is an example of Standard form. While the other two forms are Factored form and Vertex form. Example of Factored form is \[y = \left( {ax{\text{ }} + {\text{ }}c} \right)\left( {bx{\text{ }} + {\text{ }}d} \right)\] and an example of Vertex form is \[y = a\left( {x + b} \right)2 + c\]. In all the forms a, b and c are variables.

Note: Finding out the middle term factor can be difficult sometimes. So, we need to factor the constant “c” carefully and such that after addition or subtraction we get “bx”. Most importantly the plus or minus sign of the middle term should be taken seriously.