Question
State whether the following quadratic equations have two distinct real roots. Justify your answer.
$(x + 1)(x - 2) + x = 0.$
$(x + 1)(x - 2) + x = 0.$
✓
Answer
Main concept used:
Quadratic equation $ax^2 + bx + c = 0$ will have two distinct real roots if $D > 0 or b^2 - 4ac > 0.$
$(x + 1)(x - 2) + x = 0$
$\Rightarrow x^2 - 2x + x - 2 + x = 0$
$\Rightarrow x^2 - 2 = 0$
$\Rightarrow x^2 + 0x - 2 = 0$
Now, $D = b^2 - 4ac$
$= (0)^2 - 4(1)(-2) (a = 1, b = 0, c = -2)$
$\Rightarrow D = 8 > 0$
So, the given equation has two distinct real roots.
Quadratic equation $ax^2 + bx + c = 0$ will have two distinct real roots if $D > 0 or b^2 - 4ac > 0.$
$(x + 1)(x - 2) + x = 0$
$\Rightarrow x^2 - 2x + x - 2 + x = 0$
$\Rightarrow x^2 - 2 = 0$
$\Rightarrow x^2 + 0x - 2 = 0$
Now, $D = b^2 - 4ac$
$= (0)^2 - 4(1)(-2) (a = 1, b = 0, c = -2)$
$\Rightarrow D = 8 > 0$
So, the given equation has two distinct real roots.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Prove that the line segment joining the points of contact of two parallel tangents of a circle passes through its centre.
An umbrella has 8 ribs which are equally spaced (see figure). Assuming umbrella to be a flat circle of radius 45 cm, Find the area between the two consecutive ribs of the umbrella.
Find the $\text{HCF}$ and $\text{LCM}$ of $72$ and $120.$
→Find the dimensions of the prayer hall given in the below figure.
In the given figure, $AB$ and $CD$ are tangents to a circle centred at $O$ . Is $\angle BAC =\angle DCA$ ? Justify your answer.
→Find the number of coins, 1.5cm is diameter and 0.2cm thick, to be melted to form a right circular cylinder of height 10cm and diameter 4.5cm.
Very-Short and Short-Answer Questions:
If 12x + 17y = 53 and 17x + 12y =63, then find the value of (x + y).
If 12x + 17y = 53 and 17x + 12y =63, then find the value of (x + y).
Prove the following identities:
$\frac{1}{(1+\sin\theta)}+\frac{1}{(1-\sin\theta)}=2\sec^2\theta$
→$\frac{1}{(1+\sin\theta)}+\frac{1}{(1-\sin\theta)}=2\sec^2\theta$
Find the least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3
Write the first four terms of the AP, when the first term a = -1 and the common difference d = $\frac{1}{2}$
→