Question

How do you solve ${x^2} - 12x + 35 \leqslant 0$?

Answer 1

Hint: First of all, make the factors of the given expression using the method of splitting the middle term. Then, just use the fact that if $a.b \leqslant 0$, then either $a \leqslant 0$ and $b > 0$ or $a > 0$ and $b \leqslant 0$.

Complete step-by-step solution:
We are given that we are required to solve ${x^2} - 12x + 35 \leqslant 0$.
Let us assume that $f(x) = {x^2} - 12x + 35$.
We will use the method of “splitting the middle term to factorize this function.
Therefore, we can write the given equation as $f(x) = {x^2} - 5x - 7x + 35$
Taking x common from the first two terms in the function’s expression, we will then obtain the following expression:-
$ \Rightarrow f(x) = x\left( {x - 5} \right) - 7x + 35$
Taking – 7 common from the last two terms in the function’s expression, we will then obtain the following expression:-
$ \Rightarrow f(x) = x\left( {x - 5} \right) - 7\left( {x - 5} \right)$
Taking (x – 5) common from the last two terms in the function’s expression, we will then obtain the following expression:-
$ \Rightarrow f(x) = \left( {x - 5} \right)\left( {x - 7} \right)$
Now, putting this in the expression given to us, we will then obtain the following equation with us:-
$ \Rightarrow \left( {x - 5} \right)\left( {x - 7} \right) \leqslant 0$
Now, since we know that if $a.b \leqslant 0$, then either $a \leqslant 0$ and $b \geqslant 0$ or $a \geqslant 0$ and $b \leqslant 0$.
Replacing a by (x – 5) and b by (x – 7), we will then obtain the following expressions:-
Either $x - 5 \leqslant 0$ and $x - 7 \geqslant 0$ or $x - 5 \geqslant 0$ and $x - 7 \leqslant 0$ .
Either $x \leqslant 5$ and $x \geqslant 7$ or $x \geqslant 5$ and $x \leqslant 7$ .
$ \Rightarrow x \leqslant 5$ and $x \geqslant 7$ are not possible together.
Therefore, $x \geqslant 5$ and $x \leqslant 7$ which implies that $x \in \left[ {5,7} \right]$.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well. The alternate way is as follows:-
The given equation is ${x^2} - 12x + 35$.
Let us equate the given equation to 0 for once so that we can find its roots easily.
So, the equation becomes ${x^2} - 12x + 35 = 0$.
Using the quadratic formula given by if the equation is given by $a{x^2} + bx + c = 0$, its roots are given by the following equation:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Thus, we have the roots of ${x^2} - 12x + 35$ given by:
$ \Rightarrow x = \dfrac{{ - ( - 12) \pm \sqrt {{{( - 12)}^2} - 4 \times 35} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{12 \pm \sqrt {144 - 140} }}{2}$
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{12 \pm 2}}{2}$
Hence, the roots are 5 and 7.
Thus, we have the required factors and we can proceed as we did in the solution.