Question

Write the quadratic equation for which $\alpha +\beta =-4$ and $\alpha \beta =-3$
${{x}^{2}}+4x-3=0$
${{x}^{2}}+4x+3=0$
${{x}^{2}}-4x-3=0$
${{x}^{2}}-3x+4=0$

Answer 1

Hint: Here $\alpha ,\beta $ are roots so to find quadratic equation apply formula,
${{x}^{2}}-\left( \text{sum of roots} \right)x+\left( \text{product of roots} \right)=0$
Here the sum of roots is $\left( \alpha +\beta \right)$ and the product is $\alpha \beta $ .

Complete step-by-step answer:
In the question we are given $\alpha +\beta =-4$ $\alpha \beta =-3$ . Here, $\alpha ,\beta $ are the roots of the quadratic equation.
Generally quadratic equations are in the form of $a{{x}^{2}}+bx+c=0$ where a, b, c are real numbers where a is not equal to 0.
Here x represents unknown and a,b,c are known numbers where $a\ne 0$ otherwise it becomes linear as no ${{x}^{2}}$ term is there. The numbers a,b,c are coefficients of the equation and may be distinguished by calling them respectively. The quadratic coefficient, the linear coefficient and the constant or free term.
The values of x that satisfy the equation are called solutions of the equation and roots or zeros of the expression on its left hand side. A quadratic equation has at most two solutions. If there is no real solution then there are two complex solutions. If there is only one solution, one says that it is double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation of form $a{{x}^{2}}+bx+c=0$ can be factored as $a\left( x-\alpha \right)\left( x-\beta \right)=0$ where $\alpha $ and $\beta $ are solutions of x.
Because, the quadratic equation involves only one known, it is called univariate. The quadratic equation only contains powers of x that are non-negative integers and therefore, it is a polynomial equation. In particular it is a second degree polynomial equation.
A quadratic equation can also be represented as ${{x}^{2}}-\left( \text{sum of roots} \right)x+\left( \text{product of roots} \right)=0$
In the question $\alpha ,\beta $ are roots so the sum of roots given is $-4$ and product of roots are $\left( -3 \right)$ . So, according to it we get quadratic equation as,
${{x}^{2}}-\left( -4 \right)x+\left( -3 \right)=0$
So, the equation is
${{x}^{2}}+4x-3=0$
So, the correct option is ‘A’.

Note: An alternative method is also there we can solve $\alpha ,\beta $ from $\left( \alpha +\beta \right)$ and $\left( \alpha \beta \right)$ values and write the equation as $\left( x-\alpha \right)\left( x-\beta \right)$ and get desired equation but the process will be very tedious.