Question

Write $ {x^2} - 4x + 16 $ in factored form?

Answer 1

Hint: This equation is the quadratic equation. The general form of the quadratic equation is $ a{x^2} + bx + c = 0 $ . Where ‘a’ is the coefficient of $ {x^2} $ , ‘b’ is the coefficient of x and ‘c’ is the constant term.
To solve this equation, we will apply the quadratic formula for the quadratic equation.
The formula is as below:
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .
Here, $ \sqrt {{b^2} - 4ac} $ is called the discriminant. And it is denoted by $ \Delta $ .
If $ \Delta $ is greater than 0, then we will get two distinct and real roots.
If $ \Delta $ is less than 0, then we will not get real roots. In this case, we will get two complex numbers.
If $ \Delta $ is equal to 0, then we will get two equal real roots.

Complete step by step answer:
Here, the quadratic equation is
 $ \Rightarrow {x^2} - 4x + 16 $
Let us compare the above expression with $ a{x^2} + bx + c $ .
Here, we get the value of ‘a’ is 1, the value of ‘b’ is -4, and the value of ‘c’ is 16.
Now, let us find the discriminant $ \Delta $ .
 $ \Rightarrow \Delta = {b^2} - 4ac $
Let us substitute the values.
 $ \Rightarrow \Delta = {\left( { - 4} \right)^2} - 4\left( 1 \right)\left( {16} \right) $
Simplify it.
 $ \Rightarrow \Delta = 16 - 64 $
Subtract the right-hand side.
 $ \Rightarrow \Delta = - 48 $
Here, $ \Delta $ is less than 0, then we will not get real roots. We will get the complex number.
Now,
 $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
Put all the values.
 $ \Rightarrow x = \dfrac{{ - \left( { - 4} \right) \pm \sqrt { - 48} }}{{2\left( 1 \right)}} $
That is equal to
 $ \Rightarrow x = \dfrac{{4 \pm 4\sqrt 3 i}}{2} $
Let us take out 2 as a common factor from the numerator.
 $ \Rightarrow x = \dfrac{{2\left( {2 \pm 2\sqrt 3 i} \right)}}{2} $
That is equal to,
 $ \Rightarrow x = 2 \pm 2\sqrt 3 i $
Hence, the two factors are $ 2 + 2\sqrt 3 i $ and $ 2 - 2\sqrt 3 i $ .

Note: One important thing is, we can always check our work by multiplying out factors back together, and check that we have got back the original answer.
Here is a list of methods to solve quadratic equations:
• Factorization
• Completing the square
• Using graph
• Quadratic formula