Question

How do you solve $\sqrt {3x + 10} = 5 - 2x$?

Answer 1

Hint: In this question we have to solve the given polynomial , for doing this we first square both sides of the equation, then we will get a quadratic equation in the form $a{x^2} + bx + c = 0$, In the polynomial $a{x^2} + bx + c$, where "$a$", "$b$", and “$c$" are real numbers and the Quadratic Formula is derived from the process of completing the square, and is formally stated as:
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$, now by substituting the values in the formula we will get the required values for $x$.

Complete step by step solution:
Now the given quadratic equation is,
$\sqrt {3x + 10} = 5 - 2x$,
Now apply both sides of the equation, we get,
$ \Rightarrow {\left( {\sqrt {3x + 10} } \right)^2} = {\left( {5 - 2x} \right)^2}$,
Now simplifying and applying algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ on the right hand side of the equation, we get,
$ \Rightarrow 3x + 10 = 25 - 20x + 4{x^2}$,
Now taking all terms to one side we get,
$ \Rightarrow 25 - 20x + 4{x^2} - 3x - 10$,
Now simplifying we get,
$ \Rightarrow 4{x^2} - 23x + 15 = 0$,
Now the equation is in form of quadratic equation $a{x^2} + bx + c = 0$, so, using the quadratic formula, which is given by $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$,
Here$a = 4$,$b = - 23$, and $c = 15$,
Now substituting the values in the formula we get,
$ \Rightarrow x = \dfrac{{ - \left( { - 23} \right) \pm \sqrt {{{\left( { - 23} \right)}^2} - 4\left( 4 \right)\left( {15} \right)} }}{{2\left( 4 \right)}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{23 \pm \sqrt {529 - 240} }}{8}$,
Now again simplifying we get,
$ \Rightarrow x = \dfrac{{23 \pm \sqrt {289} }}{8}$,
Now taking the square root we get,
$ \Rightarrow x = \dfrac{{23 \pm 17}}{8}$,
Now we got two values for $x$, and they are,
 $ \Rightarrow x = \dfrac{{23 + 17}}{8} = \dfrac{{40}}{8} = 5$,
And second values is,
$ \Rightarrow x = \dfrac{{23 - 17}}{8} = \dfrac{6}{8} = \dfrac{3}{4}$,
So the values of $x$ are $5$and $\dfrac{3}{4}$.
$\therefore $If we solve the given equation, i.e., $\sqrt {3x + 10} = 5 - 2x$, then the values of $x$ are $5$ and $\dfrac{3}{4}$.

Note: Quadratic equation formula is a method of solving quadratic equations, but we should keep in mind that we can also solve the equation using completely the square, and we can cross check the values of $x$ by using the above formula. Also we should always convert the coefficient of${x^2} = 1$, to easily solve the equation by this method, and there are other methods to solve such kind of solutions, other method used to solve the quadratic equation is by factoring method, in this method we should obtain the solution factorising quadratic equation terms. In these types of questions, we can solve by using quadratic formula i.e., $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.