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What two numbers multiply to 90 and add to $-5$?

Last updated date: 13th Aug 2024
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Hint: We solve this problem by assuming the required two numbers as some variables and develop two equations using the information given in the question. Find one equation that two numbers multiply to 90 and another from the statement that two numbers add to $-5$ then by solving both equations we get the required numbers.

Complete step-by-step solution:
We are asked to find the two numbers that multiply to 90 and add to $-5$
Let us assume that the required two numbers as $a,b$
Now, let us take the first statement that is the two numbers multiply to 90
Now, by converting the first statement into mathematical equation then we get,
$\Rightarrow a\times b=90..........equation(i)$
Now, let us take the second statement that is the numbers add to $-5$
Now, by converting the above statement into mathematical equation then we get,
$\Rightarrow a+b=-5$
Now, let us rearrange the terms in above equation to represent one variable in terms of other variable then we get,
$\Rightarrow b=-5-a...........equation(ii)$
Now, let us substitute the value of $'b'$ from equation (ii) in equation (i) then we get
\begin{align} & \Rightarrow a\times \left( -5-a \right)=90 \\ & \Rightarrow -5a-{{a}^{2}}=90 \\ & \Rightarrow {{a}^{2}}+5a+90=0 \\ \end{align}
Here, let us find the discriminant of the above quadratic equation.
We know that the discriminant of quadratic equation of the form $a{{x}^{2}}+bx+c=0$ is given as
$\Delta ={{b}^{2}}-4ac$
By using the above discriminant formula to the quadratic equation ${{a}^{2}}+5a+90=0$ then we get,
\begin{align} & \Rightarrow \Delta ={{5}^{2}}-4\left( 1 \right)\left( 90 \right) \\ & \Rightarrow \Delta =25-360<0 \\ \end{align}
We know that if the discriminant of a quadratic equation is less than 0 then there will be imaginary roots.
Here, we can see that the discriminant of ${{a}^{2}}+5a+90=0$ is less than 0 so the roots are imaginary.
But, we know that the numbers should be real.
Therefore, we can conclude that there are no numbers that multiply to 90 and add to $-5$
$\therefore$ No two numbers multiply to 90 and add to $-5$

Note: We can give direct explanation to this problem.
We are given that two numbers multiply to 90 so that the two numbers are either positive or both negative.
We are given that the sum of two numbers is $-5$
We know that in any case the sum of two positive numbers cannot be negative.
We also know that the two negative numbers that add to $-5$ can be listed as,
(1) $0,-5$
(2) $-1,-4$
(3) $-2,-3$
Here, we can see that none of them give 90 when multiplied.
Therefore, we can conclude that there are no numbers that multiply to 90 and add to $-5$
$\therefore$ No two numbers multiply to 90 and add to $-5$