Solve x² − 2x − 1 = 0
Quadratic equation, worked out line by line the way a teacher would write it.
Answer
| Solutions | x = 1 − √2, x = 1 + √2 |
Step-by-step solution
8 steps-
1 Givenx^{2} - 2 x - 1 = 0
Solve for x.
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2 This is a quadratic in standard form ax² + bx + c = 0
Every quadratic equation can be arranged as ax² + bx + c = 0. Reading off a = 1, b = -2 and c = -1, signs included, shows which method fits: factoring, taking a square root or the quadratic formula.
a = 1,\quad b = -2,\quad c = -1 -
3 Apply the quadratic formula
The quadratic formula solves any equation ax² + bx + c = 0. It comes from completing the square on the general equation, so it always works, even when the trinomial does not factor.
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} -
4 Substitute a, b and cx = \frac{-(-2) \pm \sqrt{(-2)^{2} - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}
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5 Compute the discriminant D = b² − 4ac
D = b² − 4ac is the part under the square root in the formula. If it is positive there are two real solutions, if it is 0 there is one, and if it is negative its square root is not a real number, so there is no real solution.
D = 8Positive discriminant: two distinct real solutions.
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6 Evaluate the square root and the denominatorx = \frac{2 \pm 2 \sqrt{2}}{2}
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7 Simplifyx = 1 - \sqrt{2}\quad \text{or} \quad x = 1 + \sqrt{2}
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8 Check\begin{aligned}x = 1 - \sqrt{2}:\quad 0 = 0\quad\checkmark\\ x = 1 + \sqrt{2}:\quad 0 = 0\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 1 − √2 | 0 | 0 | ✓ |
| 1 + √2 | 0 | 0 | ✓ |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.