Question

For the given Quadratic equation \[{{\text{X}}^{\text{2}}}{\text{ - 7X - 18 = 0}}\], check the values of x from the following which satisfy the equation \[{\text{X = }}\sqrt {\text{3}} {\text{, - 2, 4}}\]

Answer 1

Hint: In solving a quadratic equation,
First of all, we find two factors whose sum equals to the coefficient of the middle term.
We find the values of \[{\text{x}}\] which make each of the factors zero.
Then, we can equate each factor to be zero and solve them.
Finally we get the required answer.

Complete step-by-step solution:
It is given that the question stated as the quadratic equation, \[{{\text{X}}^{\text{2}}}{\text{ - 7X - 18 = 0}}\]
Here we have to check if the roots are correct or not.
Now we have the first term of the given equation is \[{{\text{X}}^{\text{2}}}\] its coefficient is \[{\text{1}}\]
Also, the middle term is \[{\text{ - 7X}}\] its coefficient is \[{\text{ - 7}}\]
Also, the last term “the constant” is \[{\text{ - 18}}\]
Here we have to multiply the coefficient of the first term by the constant \[{\text{1 \times - 18 = - 18}}\]
Now we have to find the product of two factors is \[{\text{ - 18}}\] whose sum equals the coefficient of the middle term which is \[{\text{ - 7}}\]
Now we have to construct that as follows:
\[{\text{ - 18 + 1 = - 17}}\]
\[{\text{ - 2 + 9 = 7}}\]
\[{\text{ - 6 + 3 = - 3}}\]
\[{\text{ - 9 + 2 = - 7}}\]
Here we get,\[{\text{ - 9 + 2 = - 7}}\]
Rewrite the given quadratic equation as splitting the middle term using the two factors found in above, which is \[{\text{ - 9}}\] and \[{\text{2}}\]
\[ \Rightarrow {{\text{x}}^{\text{2}}}{\text{ - 9x + 2x - 18 = 0}}....\left( 1 \right)\]
Now, we have to add up the first two terms in \[\left( 1 \right)\], and pulling out like factors: \[{\text{x (x - 9)}}\]
Also, we can add up the last two terms, pulling out like factors: \[{\text{ + 2 (x - 9)}}\]
So, we get
\[ \Rightarrow {\text{(x + 2) (x - 9)}}{\kern 1pt} {\text{ = 0}}\]
This is desired factorization.
Also we can write it as, \[{\text{(x + 2) = 0}}\] and \[{\text{ (x - 9)}}{\kern 1pt} {\text{ = 0}}\]
Now we get the roots of a given quadratic equation.
\[{\text{x = - 2, x = 9}}\]

Hence, \[\sqrt {\text{3}} \] and \[{\text{4}}\] are not the roots of the given equation.

Note: A quadratic equation with real or complex coefficient has two solutions, called roots.
Real constants are polynomials of degree zero.
These two solutions may or may not be distinct; any they may or may not be real.
Factorization method can be used when the quadratic equation can be factorized into linear factors.
Given the product, if the whole product is to be zero, then any factor will be zero.
Conversely, if a product is equal to zero, then some factor of that product must be zero.