Question

Solve
\[9x - 11 + \sqrt {3x + 7} = - 31\]

Answer 1

Hint: Here we need to solve for ‘x’. First here we need to remove the radical symbol. We separate the terms which contain a radical symbol and the terms which don’t have. Then we apply squares on both sides of the equation. After simplifying we will have a quadratic equation. We can solve the quadratic equation using the formula method or by factorization method.

Complete step-by-step answer:
Given, \[9x - 11 + \sqrt {3x + 7} = - 31\].
Now Shifting the terms we have,
\[\sqrt {3x + 7} = - 31 + 11 - 9x\]
\[\sqrt {3x + 7} = - 9x - 20\]
Taking negative common we have
\[\sqrt {3x + 7} = - \left( {9x + 20} \right)\]
Now squaring on both sides we have,
\[{\left( {\sqrt {3x + 7} } \right)^2} = {\left( { - \left( {9x + 20} \right)} \right)^2}\]
We know that square and the square root will cancel out.
\[3x + 7 = {\left( {9x + 20} \right)^2}\]
Using the identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\],
\[3x + 7 = {\left( {9x} \right)^2} + {\left( {20} \right)^2} + 2\left( {9x} \right)\left( {20} \right)\]
\[3x + 7 = 81{x^2} + 400 + 360x\]
shifting all the terms in one side of the equation,
\[81{x^2} + 400 + 360x - 3x - 7 = 0\]
Adding the like terms and constants
\[81{x^2} + 357x - 393 = 0\]
Now we apply the Quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].
By comparing \[a{x^2} + bx + c = 0\], we have \[a = 81,b = 357\] and \[c = - 393\]. Substituting we have,
\[x = \dfrac{{ - 357 \pm \sqrt {{{\left( {357} \right)}^2} - 4\left( {81} \right)\left( {393} \right)} }}{{2\left( {81} \right)}}\]
\[x = \dfrac{{ - 357 \pm \sqrt {127449 - 127332} }}{{162}}\]
\[ \Rightarrow x = \dfrac{{ - 357 \pm \sqrt {117} }}{{162}}\]
Thus we have two roots
\[ \Rightarrow x = \dfrac{{ - 357 + \sqrt {117} }}{{162}}\] and \[ \Rightarrow x = \dfrac{{ - 357 - \sqrt {117} }}{{162}}\]. This is the required answer.

Note: In above we have a polynomial of degree 2, hence we have 2 solutions or roots. The highest exponent of the polynomial in a polynomial equation is called its degree. A polynomial equation has exactly as many roots as its degree. The above simplified quadratic equation can not be solved by factorization, hence we have used quadratic formulas to solve it.