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The equation $x^2 + (k - 5)x + 1 = 0$ has equal roots - Scottish Highers Maths - Question 2 - 2022

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The-equation-$x^2-+-(k---5)x-+-1-=-0$-has-equal-roots-Scottish Highers Maths-Question 2-2022.png

The equation $x^2 + (k - 5)x + 1 = 0$ has equal roots. Determine the possible values of $k$.

Worked Solution & Example Answer:The equation $x^2 + (k - 5)x + 1 = 0$ has equal roots - Scottish Highers Maths - Question 2 - 2022

Step 1

Use discriminant condition for equal roots

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Answer

For a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 to have equal roots, the discriminant must be zero. The discriminant is given by:

D=b2−4acD = b^2 - 4ac

In this case:

  • a=1a = 1
  • b=(k−5)b = (k - 5)
  • c=1c = 1

Thus, we need: D=(k−5)2−4(1)(1)=0D = (k - 5)^2 - 4(1)(1) = 0

Step 2

Apply condition and simplify

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Answer

Applying the condition: (k−5)2−4=0(k - 5)^2 - 4 = 0

Expanding this gives: (k−5)2=4(k - 5)^2 = 4

Taking the square root of both sides results in two cases:

  1. k−5=2k - 5 = 2
  2. k−5=−2k - 5 = -2

Step 3

Determine values of k

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Answer

Solving these equations:

  1. For k−5=2k - 5 = 2, we have: k=2+5=7k = 2 + 5 = 7

  2. For k−5=−2k - 5 = -2, we find: k=−2+5=3k = -2 + 5 = 3

Thus, the possible values of kk are:

k=3 or k=7k = 3 \text{ or } k = 7

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