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For which of the following values of k will (k + 10)x^2 + (k + 10)x 9

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Bunuel
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For which of the following values of k will (k + 10)x^2 + (k + 10)x 9 [#permalink] New post   20 Sep 2023, 04:30
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For which of the following values of k will \((k + 10)x^2 + (k + 10)x – 9 = 0\) have equal roots?

I. -46
II. -10
III. 46

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III
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A
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Re: For which of the following values of k will (k + 10)x^2 + (k + 10)x 9 [#permalink] New post   20 Sep 2023, 12:49
Bunuel wrote:
For which of the following values of k can \((k + 10)x^2 + (k + 10)x – 9 = 0\) have equal roots?

I. -46
II. -10
III. 46

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III


For the roots to be equal, the determinant should be zero. For a quadratic equation represented in the form \(ax^2 + bx + c \), the determinant is given by\( b^2 - 4ac\).

For the given equation, the determinant is \((k + 10)^2 - [4 (k+10) * (-9)]\)

\((k + 10)^2 +36 (k+10) = 0\)

\((k + 10)( k + 10 +36 ) = 0\)

\((k + 10)( k + 46 ) = 0\)

The value of k cannot be -10, as the quadratic equation will not hold true for k = -10.

Hence, k = -46

Option A
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Re: For which of the following values of k will (k + 10)x^2 + (k + 10)x 9 [#permalink] New post   21 Sep 2023, 11:25
Bunuel wrote:
For which of the following values of k will \((k + 10)x^2 + (k + 10)x – 9 = 0\) have equal roots?

I. -46
II. -10
III. 46

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III


\((k + 10)x^2 + (k + 10)x – 9 = 0\)

Let x1 & x2 be two roots of the eq. Since they are equal, x1 = x2 = x

x + x = \(\frac{-(k + 10)}{ (k + 10)}\)
2x = \(-1\)
x = \(\frac{-1}{ 2}\)

x*x = \(\frac{-9}{ (k + 10)}\)
\(x^2\) = \(\frac{-9}{ (k + 10)}\)

Putting value of x from above

\(\frac{ 1}{ 4}\) = \(\frac{-9}{ (k + 10)}\)
k + 10 = -36
k = -46

Ans A
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For which of the following values of k will (k + 10)x^2 + (k + 10)x 9 [#permalink] New post   22 Sep 2023, 02:17
Asked: For which of the following values of k will \((k + 10)x^2 + (k + 10)x – 9 = 0\) have equal roots?

For equal roots, discriminant should be 0.

(k+10)ˆ2 - 4*(-9)*(k+10) = 0
kˆ2 + 20k + 100 + 36k + 360 = 0
kˆ2 + 56k + 460 = 0
(k+10)(k+46) = 0
for k = -10 & k = -46, discriminant is 0, there are equal roots
But for k=-10; equation becomes -9 = 0 which is not valid

I. -46
II. -10
III. 46

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

IMO A
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For which of the following values of k will (k + 10)x^2 + (k + 10)x 9 [#permalink]
22 Sep 2023, 02:17
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