Using the quadratic formula, what are the roots of $x^2 - 4x - 1 = 0$ ?

-2 \pm \sqrt{5}

Correct answerYour answer

2 \pm \sqrt{3}

Correct answerYour answer

2 \pm 2\sqrt{5}

Correct answerYour answer

2 \pm \sqrt{5}

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

Here $a = 1$ , $b = -4$ , $c = -1$ . Discriminant $= 16 - 4(1)(-1) = 20$ .
$x = \frac{4 \pm \sqrt{20}}{2} = \frac{4 \pm 2\sqrt{5}}{2}$ .
So $x = 2 \pm \sqrt{5}$ .

Worth noting

$2 \pm \sqrt{3}$ uses $16 - 4$ (the sign of $c$ was dropped). $2 \pm 2\sqrt{5}$ divides only the 4 by 2, not the square root. $-2 \pm \sqrt{5}$ forgets that $-b = +4$ .

Concept: Quadratic formula with an irrational root
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Using the quadratic formula, what are the roots of x2−4x−1=0x^2 - 4x - 1 = 0x2−4x−1=0?

Explanation

Using the quadratic formula, what are the roots of $x^2 - 4x - 1 = 0$ ?