Question

How do you solve $ 3{x^2} - 2x - 5 = 0 $ ?

Answer 1

Hint: In this question, we need to solve the equation $ 3{x^2} - 2x - 5 = 0 $ . For splitting the middle term into two factors, we will determine the factors that multiply to give $ ac $ i.e., $ 3 \times - 5 = - 15 $ , and add to give $ b $ i.e., $ - 2 $ which is called a sum-product pattern. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor. Finally, we will equate the factors to $ 0 $ and determine the value of $ x $ .

Complete step-by-step answer:
Now, we need to solve $ 3{x^2} - 2x - 5 = 0 $ .
First, let us determine the factors of the given equation.
According to the rule to factorize,
Product= $ {x^2} $ coefficient $ \times $ constant
And, sum= $ x $ coefficient
Thus, we will find two numbers that multiply to give $ ac $ i.e., $ 3 \times - 5 = - 15 $ and add to give $ b $ i.e., $ - 2 $ ,
Here, the product is negative. So, we can say that one of the factors is negative, and then the other is positive.
Now, let’s consider the possible factors and their sum.
$
15 \times - 1 = - 15;15 + \left( { - 1} \right) = 14 \\
 5 \times - 3 = - 15;5 + \left( { - 3} \right) = 2 \\
 3 \times - 5 = - 15;3 + \left( { - 5} \right) = - 2 \\
 1 \times - 15 = - 15;1 + \left( { - 15} \right) = - 14 \;
$
From this it is clear that the factors are $ 3 $ and $ - 5 $ .
Now, by rewriting the middle term with those factors, we have,
 $ 3{x^2} + 3x - 5x - 5 = 0 $
 $ \left( {3{x^2} + 3x} \right) - \left( {5x + 5} \right) = 0 $
Factor out the greatest common factor from each group,
 $ 3x\left( {x + 1} \right) - 5\left( {x + 1} \right) = 0 $
Factor the polynomial by factoring out the greatest common factor, $ x + 1 $ ,
 $ \Rightarrow \left( {3x - 5} \right)\left( {x + 1} \right) = 0 $
Now, equate the factors separately to determine the value of $ x $ .
Therefore, $ \left( {3x - 5} \right) = 0 $ and $ \left( {x + 1} \right) = 0 $
Hence, $ x = \dfrac{5}{3} $ and $ x = - 1 $ .
So, the correct answer is “ $ x = \dfrac{5}{3} $ and $ x = - 1 $”.

Note: In this question it is important to note that this factorization method works for all quadratic equations. The standard form of the quadratic equation is $ a{x^2} + bx + c = 0 $ . It is called factoring because we find the factors. A factor is something we multiply by. There is no simple method of factorizing a quadratic expression, but with a little practice it becomes easier. Finally, equating the equation to $ 0 $ is common in all quadratic equations because we need to determine the value of the given unknown variable.