Question

How do you solve: $4{x^2} + 23x + 15 = 0$

Answer 1

Hint: In the given question, we have been asked to solve a quadratic equation which has a single variable. Quadratic equations are the polynomial expressions which have one variable with degree 2 .Since in the equation variable has degree 2, $x$will have two roots. Roots can be imaginary or real. In order to proceed with the following question we can use two approaches. We can obtain the solution either by splitting or using a quadratic formula.

Complete step by step solution:
We are given,
$ \Rightarrow 4{x^2} + 23x + 15 = 0$
To solve the splitting method, we need to find the factors of the number obtained by multiplying the coefficient of ${x^2}$ i.e. 4 by constant term i.e. 14. The number we’ll obtain is $60$. We’ll have to list down the factors of 60 whose sum is equal to the coefficient of middle term, which is 23. The factors are:
 \[
   - 60 + ( - 1) = - 61 \\
   - 30 + ( - 2) = - 32 \\
   - 20 + ( - 3) = - 23 \\
   - 15 + ( - 4) = - 19 \\
   - 12 + ( - 5) = - 17 \\
   - 10 + ( - 6) = - 16 \\
   - 6 + ( - 10) = - 16 \\
   - 5 + ( - 12) = - 17 \\
   - 4 + ( - 15) = - 19 \\
   - 3 + ( - 20) = - 23 \\
   - 2 + ( - 30) = - 32 \\
   - 1 + ( - 60) = - 61 \\
  60 + 1 = 61 \\
  30 + 2 = 32 \\
  20 + 3 = 23\;\; \\
   \]
After getting the correct factor, rewrite the equation by splitting the middle term using those two factors
 \[ \Rightarrow 4{x^2} + 3x + 20x + 15 = 0\]
Now, we have to pull out the common factors from 1st two terms
 \[ \Rightarrow x(4x + 3)\]
 and last two terms
 \[ \Rightarrow 5(4x + 3)\]
After adding up the four terms we’ll get
 \[ \Rightarrow (x + 5)(4x + 3) = 0\]
From the above equation we know,
Either,
 \[ \Rightarrow x + 5 = 0\] or \[4x + 3 = 0\]
 \[ \Rightarrow x = - 5\] or \[x = - \dfrac{3}{4}\]
Hence, these are the required values of \[x\]
So, the correct answer is “ \[ x = - 5\] or \[x = - \dfrac{3}{4}\] ”.

Note: 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 $a \ne 0$ .The factors obtained in splitting method should be a co-prime number, i.e. only one positive number should divide them both