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Find the value of $k$ for which the equation $x^2 + 4x + (k - 5) = 0$ has equal roots. - Scottish Highers Maths - Question 4 - 2017

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Find the value of $k$ for which the equation $x^2 + 4x + (k - 5) = 0$ has equal roots.

Worked Solution & Example Answer:Find the value of $k$ for which the equation $x^2 + 4x + (k - 5) = 0$ has equal roots. - Scottish Highers Maths - Question 4 - 2017

Step 1

Use the discriminant

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Answer

To determine the conditions for the equation to have equal roots, we use the discriminant condition, which states that for a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, the roots are equal if the discriminant riangle=b2−4ac riangle = b^2 - 4ac is zero.

Here, we identify:

  • a=1a = 1
  • b=4b = 4
  • c=k−5c = k - 5.

Calculating the discriminant: riangle=42−4(1)(k−5)=16−4(k−5). riangle = 4^2 - 4(1)(k - 5) = 16 - 4(k - 5). Thus, we simplify this to: riangle=16−4k+20=36−4k. riangle = 16 - 4k + 20 = 36 - 4k.

Step 2

Set the discriminant to zero

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Answer

For the equation to have equal roots, we need: 36−4k=0.36 - 4k = 0. To find kk, we rearrange the equation: 4k=364k = 36 k = rac{36}{4} k=9.k = 9.

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