Question

How many solutions does the equation \[{x^2} - 16x + 64 = 0\] have?

Answer 1

Hint: Here in this question, we have to solve the given equation, the given equation is in the form of quadratic equation. This is a quadratic equation for the variable x. By using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we can determine the solutions.

Complete step-by-step answer:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factorising or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. So the equation is written as \[{x^2} - 16x + 64 = 0\].
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\], when we compare the above equation to the general form of equation the values are as follows. a=1 b=-16 and c=64. Now substituting these values to the formula for obtaining the roots we have
\[x = \dfrac{{ - ( - 16) \pm \sqrt {{{( - 16)}^2} - 4(1)(64)} }}{{2(1)}}\]
On simplifying the terms, we have
\[ \Rightarrow x = \dfrac{{16 \pm \sqrt {256 - 256} }}{2}\]
Now subtract 256 from 256 we get
\[ \Rightarrow x = \dfrac{{16 \pm 0}}{2}\]
Hence we
\[ \Rightarrow x = \dfrac{{16}}{2}\]
On simplifying we have
\[ \Rightarrow x = 8\]
Hence we have solved the given equation and found the solutions.
Here for this question we have only one solution.
We can also solve this by using the standard algebraic formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\].
Consider the given equation \[{x^2} - 16x + 64 = 0\], when we compare to the standard algebraic formula the value of a = x and b = 8.
Substituting in the formula we have
\[ \Rightarrow {(x - 8)^2} = 0\]
Taking square root on both sides we have
\[ \Rightarrow (x - 8) = 0\]
On simplification we have
\[ \Rightarrow x = 8\]
Since the degree of the quadratic equation is 2 so we have 2 solutions. But we have only one solution.
So, the correct answer is “x = 8”.

Note: The quadratic equation can 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 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.