Question

How do you solve \[{x^2} - 9x + 14 = 0\]?

Answer 1

Hint: Here we will find the factor of the equation \[{x^2} - 9x + 14 = 0\] by using a 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.

Formula Used:
We will use quadratic formula to solve the above equation which state that to solve equation \[x = a{x^2} + bx + c\] we use the below formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\].

Complete step-by-step answer:
The equation whose factor is to be found is,
\[{x^2} - 9x + 14 = 0\]……\[\left( 1 \right)\]
The standard 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
\[a = 1,b = - 9\] and \[c = 14\]
Substituting these values in the quadratic formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we get
\[x = \dfrac{{ - \left( { - 9} \right) \pm \sqrt {{{\left( { - 9} \right)}^2} - 4\left( 1 \right)\left( {14} \right)} }}{{2 \times 1}}\]
Simplifying the expression, we get
\[ \Rightarrow x = \dfrac{{9 \pm \sqrt {81 - 56} }}{2}\]
Subtracting the terms, we get
\[ \Rightarrow x = \dfrac{{9 \pm \sqrt {25} }}{2}\]
Again simplifying the equation, we get
\[ \Rightarrow x = \dfrac{{9 \pm 5}}{2}\]
Rewriting the equation, we get
\[ \Rightarrow x = \dfrac{{9 - 5}}{2}\] and \[x = \dfrac{{9 + 5}}{2}\]
\[ \Rightarrow x = \dfrac{4}{2} = 2\] and \[x = \dfrac{{14}}{2} = 7\]

So, we get two solutions as \[x = 2\] and \[x = 7\].

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 \[a,b,c\] is known as a 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.