Question

How do you solve \[25 - {x^2} \geqslant 0\] ?

Answer 1

Hint:For solving this question we will go with the right-hand side of the equation and expand the equation by using the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ , and then we will equate the factor one by one and solve for the value of $x$ , and in this way, we will get the answer.

Formula used:
${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$
Here, $a$ and $b$ will be the variables.

Complete step by step answer:
We have the expression given as \[25 - {x^2} \geqslant 0\] . And for solving these inequalities we can write this equation as
$ \Rightarrow {5^2} - {x^2} \geqslant 0$
Now by using the formula we have discussed in the hint, we will get the equation as
$ \Rightarrow \left( {5 + x} \right)\left( {5 - x} \right) \geqslant 0$
Multiplying by negative sign both the sides, and since due to this sign of the inequalities will get changes, so the equation can be written as
$ \Rightarrow - \left( {5 + x} \right)\left( {5 - x} \right) \leqslant 0$
Now for each of the factor, we will calculate the value of $x$ one by one,
So since $5 + x \leqslant 0$
And on solving the above expression we can write it as
$ \Rightarrow x \leqslant - 5$
Similarly, for the other factor, we have the expression $5 - x \leqslant 0$
And on solving the above expression we can write it as
$ \Rightarrow x \geqslant 5$
Since in this case the solution is empty. So we will take both term positive
$5 - x \geqslant 0$
$\Rightarrow$ $x \leqslant 5$
$5 +x \geqslant 0$
$\Rightarrow$ $x \geqslant -5$
So the value of solving the expression we will get $\left[ { - 5,5} \right]$.

Hence, on solving \[25 - {x^2} \geqslant 0\] , we get $\left[ { - 5,5} \right]$.

Note: In most of the questions, solving the questions based on the inequalities we have to find the zeros for it. Like in addition and subtraction the inequality does not change but we should keep in mind that multiplying by a negative number reverses the sign of inequality. Also if we have the absolute value then also the sign will get flipped.