Which of the following equations has real roots?

3 x^{2} + 4 x + 5 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

x^{2} + x + 4 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

(x - 1) (2 x - 5) = 0

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

2 x^{2} - 3 x + 4 = 0

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is C $(x - 1) (2 x - 5) = 0$
The equation $a x^{2} + b x + c = 0$ has real roots if $b^{2} - 4 a c \geq 0.$
For A, $4^{2} - 4 \times 3 \times 5 = 16 - 60 < 0$
For B, $1^{2} - 4 \times 1 \times 1 = 1 - 4 < 0$
For C,
$(x - 1) (2 x - 5) = 0$
i.e. $2 x^{2} - 7 x + 5 = 0$
$\Rightarrow b^{2} - 4 a c = 49 - 40 = 9 > 0$
Therefore, it has real roots
Hence, option C is correct.

Suggest Corrections

Similar questions

Question 10
Which of the following equations has no real roots?
(A) $x^{2} - 4 x + 3 \sqrt{2} = 0$
(B) $x^{2} + 4 x - 3 \sqrt{2} = 0$
(C) $x^{2} - 4 x - 3 \sqrt{2} = 0$
(D) $3 x^{2} - 4 \sqrt{3} x + 4 = 0$

Question 9
Which of the following equation has two distinct real roots?
(a) $2 x^{2} - 3 \sqrt{2} x + \frac{9}{4} = 0$
(b) $x^{2} + x - 5 = 0$
(c) $x^{2} + 3 x + 2 \sqrt{2} = 0$
(d) $5 x^{2} - 3 x + 1 = 0$

Q. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i)

2 x^{2} - 3 x + 5 = 0

(ii)

3 x^{2} - 4 \sqrt{3} x + 4 = 0

(iii)

2 x^{2} - 6 x + 3 = 0

Q. Solve the following equation that has equal roots:

x^{6} - 2 x^{5} - 4 x^{4} + 12 x^{3} - 3 x^{2} - 18 x + 18 = 0

Q. Find the nature of the roots of the following quadratic equation:

3 x^{2} + 7 x + 2 = 0

{(x + 3)}^{2} + 4 x + 7 = 0

2 x^{2} - 8 x + 34 = 0

x^{2} + 2 \sqrt{3 x} - 1 = 0

Join BYJU'S Learning Program

Which of the following equations has real roots?

The correct option is C (x−1)(2x−5)=0The equation ax2+bx+c=0 has real roots if b2−4ac≥0.For A, 42−4×3×5=16−60<0For B, 12−4×1×1=1−4<0For C,(x−1)(2x−5)=0i.e. 2x2−7x+5=0⇒b2−4ac=49−40=9>0Therefore, it has real rootsHence, option C is correct.