Question

Determine the nature of roots of the following quadratic equations
(i) $x^2-11x-10=0$
(ii) $4x^2-28x+49=0$
(iii) $2x^2+5x+5=0$

Answer 1

For $a{x}^2+bx+c=0$, the discriminant, $\Delta =b^2-4ac$.

(i) Here, a=1; b=-1 and c= -10.

Now, the discriminant is $\Delta =b^2-4ac$

$=(-11{)}^2-4(1\left)\right(-10)=121+40=161$

Thus, $\Delta >0$.

Therefore, the roots are real and unequal.

(ii) Here, a=4; b=-28 and c= 49.

Now, the discriminant is $\Delta =b^2-4ac$

$=(-28{)}^2-4(4\left)\right(49)=0$

Since $\Delta =0$ 3= , the roots of the given equation are real and equal.

(iii) Here, a=2; b=5 and c= 5.

Now, the discriminant is $\Delta =b^2-4ac$

$=(5{)}^2-4(2\left)\right(5)$

$=25-40=-15$

Since $\Delta <0$, the equation has no real roots.