Find all the values of k for which the quadratic equation 2x² + kx + 8 = 0 has equal roots. Also, find the roots.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 35(b)

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section D, question 35(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(b) · 5 marks · Long answerQuadratic Equations

Find all the values of k for which the quadratic equation $2 x^{2} + k x + 8 = 0$ has equal roots. Also, find the roots.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Equal roots means the discriminant $b^{2} - 4 a c$ equals zero.

Hint 2

Here $b = k$ ; solve $k^{2} - 64 = 0$ — remember there are two values.

Hint 3

Put each value of k back in and factorise as a perfect square, or use $x = - \frac{b}{2 a}$ .

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Compare $2 x^{2} + k x + 8 = 0$ with $a x^{2} + b x + c = 0$ : $a = 2, b = k, c = 8$ . For equal roots the discriminant must be zero: $D = b^{2} - 4 a c = k^{2} - 4 (2) (8) = k^{2} - 64 = 0$ .
1 mark
$k^{2} = 64 \Rightarrow k = 8$ or $k = - 8$ , i.e. $k = \pm 8$ .
1 mark
Substituting these values gives the equations $2 x^{2} + 8 x + 8 = 0$ (for $k = 8$ ) and $2 x^{2} - 8 x + 8 = 0$ (for $k = - 8$ ).
1 mark
For $k = 8$ : $2 (x^{2} + 4 x + 4) = 0 \Rightarrow 2 (x + 2)^{2} = 0 \Rightarrow x = - 2, - 2$ .
1 mark
For $k = - 8$ : $2 (x^{2} - 4 x + 4) = 0 \Rightarrow 2 (x - 2)^{2} = 0 \Rightarrow x = 2, 2$ .
1 mark

Answer: k = ±8; roots are −2, −2 (for k = 8) and 2, 2 (for k = −8)

Where the marks are

Writes the equal-roots condition $k^{2} - 64 = 0$ (1 mark)
Obtains $k = \pm 8$ (1 mark)
Writes the two resulting equations $2 x^{2} \pm 8 x + 8 = 0$ (1 mark)
Finds root $x = - 2$ for $k = 8$ (1 mark)
Finds root $x = 2$ for $k = - 8$ (1 mark)

Revise the chapter

Chapter: Quadratic Equations (notes, formulas, where marks are lost and every past question)
Topic: Nature of Roots and Discriminant

Practise this chapterBoard-style questions from Quadratic Equations, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 430/5/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 35(b)

Hints

Step-by-step solution

Revise the chapter

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