Question

How do you factor $ {x^2} + 8x - 24? $

Answer 1

Hint: Here we use the standard quadratic equation and will find the roots of the equation comparing the given equation with the standard quadratic equations- $ a{x^2} + bx + c = 0 $ where roots will be defined as \[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\]

Complete step by step solution:
Take the given equation –
 $ {x^2} + 8x - 24 $
Arrange in the form of the standard equation –
 $ {x^2} + 8x - 24 = 0 $
Compare the above equation with the standard equation: $ a{x^2} + bx + c = 0 $
 $
   \Rightarrow a = 1 \\
   \Rightarrow b = 8 \\
   \Rightarrow c = - 24 \;
  $
Also, $ \Delta = {b^2} - 4ac $
Place the values from the given comparison
 $ \Rightarrow \Delta = {(8)^2} - 4(1)( - 24) $
Simplify the above equation –
 $ \Rightarrow \Delta = 64 + 96 $
Do Addition since negative and negative is positive. –
 $ \Rightarrow \Delta = 160 $
Take square root on both the sides of the equation –
\[ \Rightarrow \sqrt \Delta = \sqrt {160} \]
Simplify –
\[ \Rightarrow \sqrt \Delta = 4\sqrt {10} \]
The above expression can not be further reduced since the number on the right is not the perfect square.
Now, roots of the given equation can be expressed as –
\[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\]
Place values in the above equation –
\[x = \dfrac{{ - (8) \pm 4\sqrt {10} }}{2}\]
 Simplify the above equation –
\[x = \dfrac{{2( - 4 \pm 2\sqrt {10} )}}{2}\]
Common factors from the numerator and the denominator cancel each other.
Therefore \[x = - 4 + 2\sqrt {10} \] or \[x = - 4 - 2\sqrt {10} \]
This is the required solution.
So, the correct answer is “\[x = - 4 + 2\sqrt {10} \] or \[x = - 4 - 2\sqrt {10} \]”.

Note: Be careful regarding the sign convention. Always remember that the square of negative number or the positive number is always positive. Also, product of two negative numbers is always positive whereas, product of one positive and one negative number gives us the negative number. Perfect square number is the square of an integer, simply it is the product of the same integer with itself. For example - $ 25{\text{ = 5 }} \times {\text{ 5, 25 = }}{{\text{5}}^2} $ , generally it is denoted by n to the power two i.e. $ {n^2} $ .