Question

Obtain the quadratic equation if roots are \[ - 3,\, - 7\].

Answer 1

Hint: Here in this question, we have to make a quadratic equation by the given data. Any equation which consists of a variable with the square on it is known as a quadratic equation. It is represented by this standard form \[a{x^2} + bx + c = 0\]. We know that if we are given the roots of a quadratic equation as A, B then we can find that Quadratic equation by using the formula $x^2-{A+B}x+AB=0$.

Complete step-by-step solution:
Firstly, consider the given data as \[A\] and \[B\] which are the roots of a Quadratic equation.
Let the given data \[ - 3, - 7\] be \[A\] and \[B\] Respectively.
So, we will make an equation out of the given data, and we get:
\[A\]=\[ - 3\], \[B\]=\[ - 7\]
Now, as we know the relation between roots and coefficient of a quadratic equation, now adding the values for making a quadratic equation
\[A + B = - 3 - 7 = - 10\]
\[ \Rightarrow AB = \left( { - 3} \right)\left( { - 7} \right) = 21\]
Now, we will put them together in order to make an equation, and we get:
\[ \Rightarrow {x^2} - \left( {A + B} \right)x + AB = 0\]
\[ \Rightarrow {x^2} - \left( { - 10x} \right) + 21 = 0\]
\[ \Rightarrow {x^2} + 10x + 21 = 0\],
As the variable has a potential power of two or we can call it a square it is now considered as the quadratic equation.

Note: One thing that should not remain unnoticed is the value of ‘a’ can’t be zero in the standard form of the equation. Similar to the quadratic equation, there are Linear equations as well. The difference they have is the power of the variable is less than two, which makes it a linear number, hence it is called a linear equation.