Question

How do you solve \[{x^2} - 20x + 100 = 36\]?

Answer 1

Hint: Here in this question, we have to solve the given quadratic equation and find the roots for the variable x. we can solve above equation by factoring or by the considering or applying the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]or we can also use sum product rule. Hence, we obtain the required result.

Complete step-by-step solution:
The equation is in the form of a quadratic equation. The degree of the equation is 2. Therefore, we obtain two values after simplifying. Here we solve the equation by using formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
In general, the quadratic equation will be in the form of \[a{x^2} + bx + c\], where a, b, c are constants and the x is variable.
Now consider the equation \[{x^2} - 20x + 100 = 36\]
Take 36 to the LHS, the equation is written as
\[{x^2} - 20x + 100 - 36 = 0\]
On simplifying we get
\[{x^2} - 20x + 64\]
When we compare the given equation to the general form of equation we have a = 1, b = -20 and c = 64.
Substituting these values in the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] we get
\[ \Rightarrow x = \dfrac{{ - ( - 20) \pm \sqrt {{{(20)}^2} - 4(1)(64)} }}{{2(1)}}\]
On simplifying we get
\[ \Rightarrow x = \dfrac{{20 \pm \sqrt {400 - 256} }}{2}\]
On further simplification we get
\[ \Rightarrow x = \dfrac{{20 \pm \sqrt { - 144} }}{2}\]
The number 169 is a perfect square. So the square root 169 is 12. Since it contains negative value in the square root we consider it as “i” and the above equation can be written as
\[ \Rightarrow x = \dfrac{{20 \pm 12i}}{2}\]
We write the two terms separately, so we have
\[ \Rightarrow x = \dfrac{{20 + 12i}}{2}\] and \[x = \dfrac{{20 - 12i}}{2}\]
Take 2 as common in the numerator and on simplifying the above terms we have
\[ \Rightarrow x = \dfrac{{2(10 + 6i)}}{2}\] and \[x = \dfrac{{2(10 - 6i)}}{2}\]
On further simplification we have
\[ \Rightarrow x = 10 + 6i\] and \[x = 10 - 6i\]
Hence we have solved the given equation and found the roots for the equation.

Note: The equation is a quadratic equation. This problem can be solved by using the sum product rule. This defines as for the general quadratic equation \[a{x^2} + bx + c\], the product of \[a{x^2}\] and c is equal to the sum of bx of the equation or we can solve the equation by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. Sometimes the factorisation method is not applicable to solve the quadratic equation.