Question

Write whether the following statements are true or false.

Justify your answers.

If the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of $x$ term is zero, then the quadratic equation has no real roots.

• A. True
• B. False

Answer 1

The correct option is A True

Roots of a quadratic equation:

1. The root of an equation is a value of the variable for which the equation is satisfied.
2. The number of roots an equation has is determined by the degree of the equation.
3. The roots may be real and distinct, they may be equal or they may imaginary.
4. The degree of the equation is the highest power of the variable in the equation.
5. A quadratic equation is an equation of degree $2$.

If $a{x}^2+bx+c=0$ is a quadratic equation, then its roots are given as,
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The value of $b^2-4ac$ determines the nature of the roots of a quadratic equation.

If,

1. $b^2-4ac>0$, then the equation has real and distinct roots.
2. $b^2-4ac=0$, then the equation has real and equal roots.
3. $b^2-4ac<0$, then the equation has complex roots.

For this reason, it is called the discriminant.

So for the roots to be complex, we need that $b^2-4ac<0$.
If the coefficient of the $x$ term i.e., $b$ is zero, then the above relation becomes $-4ac<0$
If we have that the coefficient of $x^2$ i.e., $a$, and the constant term i.e., $c$, have same signs, then $ac>0$ i. e, $ac$ will have positive sign.
Then, $-4ac<0$, since negative sign times positive sign is negative sign, which would imply that the discriminant is less than zero.
Then, $b^2-4ac<0$ will be satisfied and the equation will have complex roots.

Thus, if the coefficient of $x^2$ and the constant term have the same sign and if the coefficient of $x$ term is zero, then the quadratic equation has no real roots.

Hence, the given statement is true.