You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard X

Mathematics

Solving a Quadratic Equation by Completing the Square

The number of...

Question

The number of values of $k$ for which ${x^{2} - (k - 2) x + k^{2}} + {x^{2} + k x + (2 k - 1)}$ is a perfect square is/are

1

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

2

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

0

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

none of these

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

Open in App

Solution

The correct option is D $2$
Consider $p (x) = {x^{2} - (k - 2) x + k^{2}} + {x^{2} + k x + (2 k - 1)}$
$\Rightarrow p (x) = 2 x^{2} + 2 x + k^{2} + 2 k - 1$
$p (x)$ is perfect square when roots of the equation $p (x) = 0$ are equal
Therefore, $D = b^{2} - 4 a c = 4 - 8 (k^{2} + 2 k - 1) = 0$
$\Rightarrow 2 k^{2} - 4 k - 3 = 0$
Therefore, number of values of $k$ are 2.
Ans: B

Suggest Corrections

Similar questions

x^{2} + x + 1 + 2 k (x^{2} - x - 1) = 0

is perfect square for how many values of

k

Q. For what values of

k, (4 - k) x^{2} + (2 k + 4) x + (8 k + 1) = 0

, is a perfect square.

Q. The value of k for which the roots are real and equal of the following equation

x^{2}

- 2(k + 1)x +

k^{2}

= 0 is k =

- \frac{1}{2}

Q. Find the values of k for which the roots are real and equal in each of the following equations:

(i)

k x^{2} + 4 x + 1 = 0

(ii)

k x^{2} - 2 \sqrt{5} x + 4 = 0

(iii)

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

(iv)

4 x^{2} + k x + 9 = 0

(v)

2 k x^{2} - 40 x + 25 = 0

(vi)

9 x^{2} - 24 x + k = 0

(vii)

4 x^{2} - 3 k x + 1 = 0

(viii)

x^{2} - 2 (5 + 2 k) x + 3 (7 + 10 k) = 0

(ix)

(3 k + 1) x^{2} + 2 (k + 1) x + k = 0

(x)

k x^{2} + k x + 1 = - 4 x^{2} - x

(xi)

(k + 1) x^{2} + 2 (k + 3) x + (k + 8) = 0

(xii)

x^{2} - 2 k x + 7 k - 12 = 0

(xiii)

(k + 1) x^{2} - 2 (3 k + 1) x + 8 k + 1 = 0

(xiv)

(2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0

(xvii)

4 x^{2} - 2 (k + 1) x + (k + 4) = 0

(xviii)

x^{2} - 2 (k + 1) x + k^{2} = 0

(xix)

k^{2} x^{2} - 2 (2 k - 1) x + 4 = 0

(xx)

(k + 1) x^{2} - 2 (k - 1) x + 1 = 0

(xxi)

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

(xxii)

k x (x - 2) + 6 = 0

(xxiii)

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

(xxiv)

k x (x - 2 \sqrt{5}) + 10 = 0

(xxv)

p x (x - 3) + 9 = 0

(xxvi)

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

Q. Number of values of

k

so that the equations

x^{2} + k x + (k + 2) = 0 and x^{2} + (1 - k) x + 3 - k = 0

have exactly one common root, is -

Join BYJU'S Learning Program

The number of values of k for which {x2−(k−2)x+k2}+{x2+kx+(2k−1)} is a perfect square is/are

The correct option is D 2Consider p(x)={x2−(k−2)x+k2}+{x2+kx+(2k−1)}⇒p(x)=2x2+2x+k2+2k−1p(x) is perfect square when roots of the equation p(x)=0 are equalTherefore, D=b2−4ac=4−8(k2+2k−1)=0⇒2k2−4k−3=0Therefore, number of values of k are 2.Ans: B

The number of values of $k$ for which ${x^{2} - (k - 2) x + k^{2}} + {x^{2} + k x + (2 k - 1)}$ is a perfect square is/are