Question

How do you solve $ {x^2} + 3x - 40 = 0 $ ?

Answer 1

Hint: In this question, we need to solve the equation $ {x^2} + 3x - 40 = 0 $ . For splitting the middle term into two factors, we will determine the factors that multiply to give $ ac $ i.e., $ 1 \times - 40 = - 40 $ , and add to give $ b $ i.e., $ 3 $ which is called 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 $ {x^2} + 3x - 40 = 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., $ 1 \times - 40 = - 40 $ and add to give $ b $ i.e., $ 3 $ ,
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.
$
  40 \times - 1 = - 40;40 + \left( { - 1} \right) = 39 \\
  20 \times - 2 = - 40;20 + \left( { - 2} \right) = 18 \\
  10 \times - 4 = - 40;10 + \left( { - 4} \right) = 6 \\
  8 \times - 5 = - 40;8 + \left( { - 5} \right) = 3 \\
  5 \times - 8 = - 40;5 + \left( { - 8} \right) = - 3 \\
  4 \times - 10 = - 18;4 + \left( { - 10} \right) = - 6 \\
  2 \times - 20 = - 40;2 + \left( { - 20} \right) = - 18 \\
  1 \times - 40 = - 40;1 + \left( { - 40} \right) = - 39 \;
$
From this it is clear that the factors are $ 8 $ and $ - 5 $ .
Now, by rewriting the middle term with those factors, we have,
  $ {x^2} + 8x - 5x - 40 = 0 $
  $ \left( {{x^2} + 8x} \right) - \left( {5x - 40} \right) = 0 $
  $ x\left( {x + 8} \right) - 5\left( {x + 8} \right) = 0 $
Factor out the greatest common factor from each group,
  $ x\left( {x + 8} \right) - 5\left( {x + 8} \right) = 0 $
Factor the polynomial by factoring out the greatest common factor, $ x + 8 $ ,
  $ \Rightarrow \left( {x - 5} \right)\left( {x + 8} \right) = 0 $
Now, equate the factors separately to determine the value of $ x $ .
Therefore, $ \left( {x - 5} \right) = 0 $ and $ \left( {x + 8} \right) = 0 $
Hence, $ x = 5 $ and $ x = - 8 $ .
So, the correct answer is “ $ x = 5 $ and $ x = - 8 $ ”.

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 factoring 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.