Question

How do you factor \[{x^2} - 45x + 450\]?

Answer 1

Hint: Here we will find the factor of the equation \[{x^2} - 45x + 450\] by using the quadratic formula. First, we will compare the given equation with the general form of the quadratic equation to get the value of all the coefficients. Then we will substitute these values in the formula and solve it further to get the solution to the equation. Using the solution, we will find the required factor.

Formula Used:
We will use the quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step-by-step answer:
The equation whose factor is to be found is,
\[{x^2} - 45x + 450\]……\[\left( 1 \right)\]
The general form of quadratic equation is given by
\[x = a{x^2} + bx + c\]…..\[\left( 2 \right)\]
On Comparing equation \[\left( 1 \right)\] and \[\left( 2 \right)\], we get
\[a = 1,b = - 45\] and \[c = 450\]
Substituting these values in the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we get
\[x = \dfrac{{ - \left( { - 45} \right) \pm \sqrt {{{\left( { - 45} \right)}^2} - 4\left( 1 \right)\left( {450} \right)} }}{{2 \times 1}}\]
Simplifying the equation, we get
\[ \Rightarrow x = \dfrac{{45 \pm \sqrt {2025 - 1800} }}{2}\]
Subtracting the terms, we get
\[ \Rightarrow x = \dfrac{{45 \pm \sqrt {225} }}{2}\]
Taking square root of 225, we get
\[ \Rightarrow x = \dfrac{{45 \pm 15}}{2}\]
Rewriting the above equation, we get
\[ \Rightarrow x = \dfrac{{45 - 15}}{2}\] or \[x = \dfrac{{45 + 15}}{2}\]
Adding and subtracting the terms, we get
\[ \Rightarrow x = \dfrac{{30}}{2} = 15\] or \[x = \dfrac{{60}}{2} = 30\]
So, we got our two solutions as \[x = 15\] and \[x = 30\].
Therefore, the factors of the equation are \[\left( {x - 15} \right)\] and \[\left( {x - 30} \right)\].

Note: An equation is said to be quadratic if it can be written as in standard form \[a{x^2} + bx + c = 0\] where x is our unknown variable and \[a,b,c\] are known as quadratic coefficient, linear coefficient and constant term respectively. A quadratic equation has two solutions because its highest degree is 2. There are many methods to find the factors of a quadratic equation such as by completing the square, Factoring by Inspection method, Discriminant and Geometrical Interpretation. The value of roots can be positive, negative and even complex also and these roots or the value that satisfy the equation are known as solutions of the equation.