Question

Solve the following quadratic equation: \[{x^2} + 7x - 98 = 0\]

Answer 1

Hint: We will solve the equation by two methods: middle term method and Sreedhar Acharya’s method. Quadratic factorization using splitting of the middle term: In this method splitting of the middle term into two factors. That is Middle Term which is \[x\] term is the sum of two factors and product equal to last term. We will factorize 98 and write the middle term \[7x\] by the help of the factors of 98.

Formula used: Sreedhar Acharya’s method: It states that, if \[a{x^2} + bx + c = 0\],
Where, \[a \ne 0\], the roots will be \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step answer:
It is given that: \[{x^2} + 7x - 98 = 0\], we have to find the solution of \[{x^2} + 7x - 98 = 0\].
It is a polynomial of second degree of one variable. It is known as a quadratic function. We will try to solve the equation by middle term method.
And we know that, if the product of two or more terms is zero, each of the terms is individually zero.
We will factorize 98 and try to write the middle term \[7x\] by the help of the factors of 98.
Simplifying we get,
$\Rightarrow$\[{x^2} + (14 - 7)x - 98 = 0\]
Multiply the terms x into brackets,
$\Rightarrow$\[{x^2} + 14x - 7x - 98 = 0\]
Factoring we get,
$\Rightarrow$\[x(x + 14) - 7(x + 14) = 0\]
Hence,
$\Rightarrow$\[(x + 14)(x - 7) = 0\]
We know that, if the product of two or more terms is zero, each of the terms is individually zero.
So, \[(x + 14) = 0\] and \[(x - 7) = 0\]
Solving we get,
\[x = - 14\] and \[x = 7\].

$\therefore $ The value of \[x\] is \[ - 14,7\].

Note: We can solve the equation by Sreedhar Acharya’s method.
It states that, if \[a{x^2} + bx + c = 0\], where, \[a \ne 0\], the roots will be \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Roots are the solution of the given equation.
Substitute, \[a = 1,b = 7,c = - 98\] we get,
The roots are,
$\Rightarrow$\[x = \dfrac{{ - 7 \pm \sqrt {{7^2} - 4 \times 1 \times ( - 98)} }}{{2 \times 1}}\]
Solving we get,
$\Rightarrow$\[x = \dfrac{{ - 7 \pm \sqrt {{7^2} - 4 \times 1 \times ( - 98)} }}{{2 \times 1}}\]
Solving we get,
$\Rightarrow$\[x = \dfrac{{ - 7 + 21}}{2} = 7\] and \[x = \dfrac{{ - 7 - 21}}{2} = - 14\]
Hence, the value of \[x\] is \[ - 14,7\].