Question

Solve: \[27{x^2} - 10x + 1 = 0\]

Answer 1

Hint:
Here, we will solve for the value of the variable. We will compare the given equation to the standard form of the quadratic equation to find the coefficients and constants. Then we will use these values and substitute in the quadratic formula. We will solve it further to get the value of the variable. A quadratic equation is an equation of a variable with the highest degree of 2.

Formula Used:
Quadratic formula to solve the quadratic equation is given by \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] where \[a,b,c\] be the coefficient of \[{x^2}\], coefficient of \[x\] and the constant term respectively.

Complete Step by Step Solution:
We are given with a quadratic equation \[27{x^2} - 10x + 1 = 0\].
The quadratic equation is of the form \[a{x^2} + bx + c = 0\].
By comparing the given quadratic equation with the general quadratic equation, we get
\[a = 27\]
\[b = - 10\]
\[c = 1\]
Substituting \[a = 27\], \[b = - 10\] and \[b = - 10\] in the quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we get
 \[x = \dfrac{{ - \left( { - 10} \right) \pm \sqrt {{{\left( { - 10} \right)}^2} - 4\left( {27} \right)\left( 1 \right)} }}{{2\left( {27} \right)}}\]
We know that when a negative integer is multiplied by a negative integer, then the resulting integer would be positive. Thus, we get
\[ \Rightarrow x = \dfrac{{\left( {10} \right) \pm \sqrt {100 - 108} }}{{54}}\]
By subtracting the terms, we get
\[ \Rightarrow x = \dfrac{{\left( {10} \right) \pm \sqrt { - 8} }}{{54}}\]
We know that the square root of a negative number will result in a complex number i.e., \[{i^2} = - 1\] .
\[ \Rightarrow x = \dfrac{{\left( {10} \right) \pm \sqrt {4 \times 2 \times - 1} }}{{54}}\]
\[ \Rightarrow x = \dfrac{{\left( {10} \right) \pm 2i\sqrt 2 }}{{54}}\]
By taking out the common factors, we get
\[ \Rightarrow x = \dfrac{{2\left( {5 \pm i\sqrt 2 } \right)}}{{54}}\]
Dividing numerator and denominator by 2, we get
\[ \Rightarrow x = \dfrac{{\left( {5 \pm i\sqrt 2 } \right)}}{{27}}\]
By separating the terms, we get
\[ \Rightarrow x = \dfrac{{\left( {5 + i\sqrt 2 } \right)}}{{27}}\] and \[x = \dfrac{{\left( {5 - i\sqrt 2 } \right)}}{{27}}\]
\[ \Rightarrow x = \dfrac{5}{{27}} + \dfrac{{i\sqrt 2 }}{{27}}\] and \[x = \dfrac{5}{{27}} - \dfrac{{i\sqrt 2 }}{{27}}\]

Therefore, the value of \[x\]is \[\dfrac{5}{{27}} + \dfrac{{i\sqrt 2 }}{{27}}\] and \[\dfrac{5}{{27}} - \dfrac{{i\sqrt 2 }}{{27}}\].

Note:
We know that we can solve the quadratic equation by using any of the four methods. Some quadratic equations cannot be solved by using the factorization method and square root method. But we can solve any quadratic equation by using the method of quadratic formula. We should be careful that the quadratic equation should be arranged in the standard form. Also, we have both the positive and negative signs in the formula, so the solutions for the equations would be according to the signs.