Question
Question: Solve for ‘x’, \({{9}^{x+2}}-{{6.3}^{x+2}}+1=0\)?...
Solve for ‘x’, 9x+2−6.3x+2+1=0?
Solution
We start solving the problem by applying the law of exponents am+n=am.an in the given equation. We then apply the law of exponents (am)n=(an)m and assume the 3x=y to get a quadratic equation in ‘y’. We then find the roots of the obtained quadratic equation in ‘y’ and equate it to 3x. We then make use of laws of exponents ax1=a−x and if am=an then m=n to get the required value of ‘x’.
Complete step by step answer:
According to the problem we need to find the value of x which satisfies the equation 9x+2−6.3x+1+1=0.
So, we have 9x+2−6.3x+1+1=0.
From the laws of exponents, we know that am+n=am.an.
So, we get 92.9x−6.31.3x+1=0.
⇒81.9x−6.3.3x+1=0.
⇒81.(32)x−18.3x+1=0.
From the law of exponents, we know that (am)n=(an)m.
⇒81.(3x)2−18.3x+1=0.
Let us assume 3x=y ---(1). So, we get 81y2−18y+1=0.
We can see that 81y2−18y+1=0 resembles a quadratic equation ax2+bx+c=0. Let us factorize it and find the roots.
⇒81y2−9y−9y+1=0.
⇒9y(9y−1)−1(9y−1)=0.
⇒(9y−1)(9y−1)=0.
⇒(9y−1)2=0.
⇒9y−1=0.
⇒9y=1.
⇒y=91. Let us substitute this in equation (1).
⇒3x=91.
⇒3x=321.
From the law of exponents, we know that ax1=a−x.
⇒3x=3−2.
From the law of exponents, we know that if am=an then m=n.
∴x=−2.
So, we have found the value of ‘x’ as –2.
Note: Whenever we get this type of problems, we try to assume a variable for the term with the independent variable in exponent to avoid confusion while solving this problem. We should not make mistakes while applying the laws of exponents in this problem. We can also solve for the roots of the quadratic equation 81y2−18y+1=0 by applying 2a−b±b2−4ac. Similarly, we can expect problems to find the value of 5x+2 after finding the value of ‘x’.