Question

Applying zero product rule for the equation \[{{x}^{2}}-ax-30=0\] is \[x=10\], then the value of a is
(a) 6
(b) 7
(c) 8
(d) 9

Answer 1

Hint: Assume that the given equation can be factorized as \[\left( x-u \right)\left( x-v \right)=0\]. Use zero product rule to conclude that \[x=u\] or \[x=v\]. Simplify the equation \[\left( x-u \right)\left( x-v \right)=0\] and compare it with \[{{x}^{2}}-ax-30=0\] to find the value of u and v. Substitute the values of u and v in the equation \[a=u+v\] to get the value of a.

Complete step-by-step answer:
We have the equation \[{{x}^{2}}-ax-30=0\]. We know that one of its roots is \[x=10\]. We have to calculate the value of ‘a’.
Let’s assume that we can factorize \[{{x}^{2}}-ax-30=0\] as \[\left( x-u \right)\left( x-v \right)=0\].
So, the roots of the equation \[{{x}^{2}}-ax-30=0\] and \[\left( x-u \right)\left( x-v \right)=0\] are the same.
Using zero product rule for \[\left( x-u \right)\left( x-v \right)=0\], we have \[x-u=0\] or \[x-v=0\].
Thus, the roots of the equation \[\left( x-u \right)\left( x-v \right)=0\] are \[x=u\] or \[x=v\].
We know that one of the roots of the equation \[{{x}^{2}}-ax-30=0\] is \[x=10\].
So, one of the roots of \[\left( x-u \right)\left( x-v \right)=0\] will be \[x=10\].
Let’s say \[u=10\].
We will now find the value of ‘v’.
On simplifying the equation \[\left( x-u \right)\left( x-v \right)=0\], we have \[{{x}^{2}}-\left( u+v \right)x+uv=0\]. This equation is the same as \[{{x}^{2}}-ax-30=0\].
On comparing both the equations, we have \[-a=-\left( u+v \right)\] and \[-30=uv\].
We know that \[u=10\].
Thus, we have \[-30=10v\].
So, we have \[v=\dfrac{-30}{10}=-3\].
Thus, we have \[u=10,v=-3\].
We know that \[-a=-\left( u+v \right)\].
Substituting the values in the above equation, we have \[-a=-\left( u+v \right)=-\left( 10-3 \right)\Rightarrow a=7\].
Hence, the value of a is \[a=7\], which is option (b).

Note: We can check if the calculated value of ‘a’ is correct or not by substituting the value of a in the equation and then finding the roots of the equation and check if they match with the above calculated roots or not. Zero product rule says that the product of two non – zero elements is zero. Or, equivalently, if the product of two numbers is zero, then one of them will be equal to zero.