Question

How do you solve $ 25{x^2} - 24x - 9 = - 7{x^2} + 12x - 18 $ ?

Answer 1

Hint: In the given question, we have been asked to unravel a linear equation with a single variable. Since in the equation variable has exponent 2, $ x $ will have two values. In order to proceed with the subsequent question we need to rearrange the following equation by writing the like terms along i.e. all the variables together and constants together. In order to proceed with the following question we can use two approaches. We can acquire the solution either by splitting or using a quadratic formula.

Complete step by step solution:
We are given,
 $ 25{x^2} - 24x - 9 = - 7{x^2} + 12x - 18 $
Firstly, we have to bring the terms of the same form together and rewrite the equation to solve the question. As we know that two numbers can solely be operated if they are of same type and coefficients of same power of variable can solely be operated. Also, we need to keep in mind that sign of the term becomes opposite when sent on the other side of “ $ = $ ” sign
 $ \Rightarrow 25{x^2} + 7{x^2} - 24x - 12x - 9 + 18 = 0 $
 $ \Rightarrow 32{x^2} - 36x + 9 = 0 $
To solve by quadratic formula, we’ll put the values in the formula
 $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
Where,
 $
  a = 32 \\
  b = - 36 \\
  c = 9 \;
  $
 $ \Rightarrow \dfrac{{ - ( - 36) \pm \sqrt {{{( - 36)}^2} - 4 \times 32 \times 9} }}{{2 \times 32}} $
 $ \Rightarrow \dfrac{{36 \pm \sqrt {1296 - 1152} }}{{64}} $
 $ \Rightarrow \dfrac{{36 \pm \sqrt {144} }}{{64}} $
 $ \Rightarrow \dfrac{{36 \pm 12}}{{64}} $
Now we’ll have to solve two terms separately- one with plus and one with minus
The one with plus-
 $ \Rightarrow \dfrac{{36 + 12}}{{64}} $
 $ \Rightarrow \dfrac{{48}}{{64}} $
\[ \Rightarrow \dfrac{3}{4}\]
The one with minus
 $ \Rightarrow \dfrac{{36 - 12}}{{64}} $
 $ \Rightarrow \dfrac{{24}}{{64}} $
 $ \Rightarrow \dfrac{3}{8} $
The two values of $ x $ are $ \dfrac{3}{4}\;and\;\dfrac{3}{8} $ .
So, the correct answer is “ $ \dfrac{3}{4}\;and\;\dfrac{3}{8} $ ”.

Note: Operations like addition, subtraction, multiplication and division can solely be performed on terms which are of similar form. Before solving any question of quadratic equation, ensure that the equation is of the form $ a{x^2} + bx + c = 0 $ , and if it is not, then convert it in this form, where $ a,b,c \in R $ and $ a \ne 0 $ .