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Question: Solve the given expression: \[(x - 5)(x - 6) = \dfrac{{25}}{{{{24}^2}}}\]...

Solve the given expression: (x−5)(x−6)=25242(x - 5)(x - 6) = \dfrac{{25}}{{{{24}^2}}}

Explanation

Solution

The given question need to be solved by expanding the expression and then solve for the value of the variable, here after expanding we will get the quadratic equation which need to be solved with mid term splitting rule if possible and if not possible then we can solve by sridharacharya rule.

Formula Used:
D=b2−4acD = \sqrt {{b^2} - 4ac} formula for the discriminant in the quadratic equation, for any general equation.
x=−b±b2−4ac2ax = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} formulae for finding the roots of the quadratic equation, it may be positive, negative real or complex, depending upon the values of the equation.

Complete step by step solution:
The given question need to solve for the variable in the expression after expanding the expression:
The given question is (x−5)(x−6)=25242(x - 5)(x - 6) = \dfrac{{25}}{{{{24}^2}}}
Here after expanding the expression we get:

⇒x(x−6)−5(x−6)=25242 ⇒x2−6x−5x+30=25242 ⇒x2−11x+30=25576=0.043 ⇒x2−11x+29.957=0  \Rightarrow x(x - 6) - 5(x - 6) = \dfrac{{25}}{{{{24}^2}}} \\\ \Rightarrow {x^2} - 6x - 5x + 30 = \dfrac{{25}}{{{{24}^2}}} \\\ \Rightarrow {x^2} - 11x + 30 = \dfrac{{25}}{{576}} = 0.043 \\\ \Rightarrow {x^2} - 11x + 29.957 = 0 \\\

Here mid term splitting rule is not possible hence we have to get to the sridharacharya rule to solve the solution for the variable, on solving we get:

⇒x2−11x+29.957=0 ⇒D=b2−4ac=(−11)2−4(1)(29.957)=121−119.82=1.08 ⇒x=−b±b2−4ac2a=−(11)±1.082(1)=−5.04,−6.04  \Rightarrow {x^2} - 11x + 29.957 = 0 \\\ \Rightarrow D = \sqrt {{b^2} - 4ac} = \sqrt {{{( - 11)}^2} - 4(1)(29.957)} = \sqrt {121 - 119.82} = 1.08 \\\ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - (11) \pm 1.08}}{{2(1)}} = - 5.04, - 6.04 \\\

Here we got the values for the variable in our expression, both the values are negative, but real values, to check if the values are correct or not we can put in the main equation and get the result for the equality of left hand side to the right hand side of the expression.

Note: The given question needs to be simplified for getting the solution for the variable, and here we have to use the sridharacharya rule because mid term splitting is not possible to get the roots of the equation. This question needs to be solved by this method only.