Question

What is the minimum height of any point on the curve $$y = {x^2} - 4x + 6$$ above the x-axis?
A.1
B.2
C.3
D.4

Answer 1

Hint: Here in this question, we need to find the minimum height of point on the given equation of curve. For this, we have to find the factors or write factored for of given equation of curve by using a algebraic identity $${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\left( {a - b} \right)\left( {a + b} \right)$$ and on further simplification, we get the required solution.

Complete answer:
Factored form is defined as the simplest algebraic expression in which no common factors remain. Finding the factored form is useful in solving linear equations. Factored form may be a product of greatest common factors or the difference of squares.
Consider the given algebraic expression:
$$ \Rightarrow \,\,\,y = {x^2} - 4x + 6$$------(1)
Add and subtract 2 on the LHS of the above equation, then we have
$$ \Rightarrow \,\,\,y = {x^2} - 4x + 6 + 2 - 2$$
On simplification, we get
$$ \Rightarrow \,\,\,y = {x^2} - 4x + 4 + 2$$
Or
$$ \Rightarrow \,\,\,y = {x^2} - 4x + {2^2} + 2$$-------(2)
Now, the algebraic equation $$y = {x^2} - 4x + {2^2}$$ its look similar as algebraic identity $${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\left( {a - b} \right)\left( {a + b} \right)$$
[Algebraic Identities The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials.]
here a=x and b=2
Then equation (2) becomes:
$$ \Rightarrow \,\,\,y = {\left( {x - 2} \right)^2} + 2$$
Now, the factor
$${\left( {x - 2} \right)^2} \geqslant 0$$ for all real x.
Add 2 on both side, then we get
$${\left( {x - 2} \right)^2} + 2 \geqslant 2$$ for all real x.
Hence, the minimum height of the curve $$y = {x^2} - 4x + 6$$ is 2 above the x-axis.
Therefore, option (B) is the correct answer.

Note:
The quadratic equation can also be solved by using the factorisation method and we also find the roots by using the formula $$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$$. While factorising we use the sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.