Solve x² - 7x + 1 = 0
The solutions of x² - 7x + 1 = 0 are x = (7 + 3√5)/2 or x = (7 - 3√5)/2, that is x ≈ 6.854102 and x ≈ 0.14589803.
Change a, b or c (whole numbers, a not 0) and the exact solutions follow.
How to work it out
- 1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -7, c = 1.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is positive, so there are two different real solutions.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ 6.854102 and x ≈ 0.14589803.
- 7
Check numerically: for x ≈ 6.854102 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 - 7x + 1 = 0\ \Rightarrow\ x = \frac{7 + 3\sqrt{5}}{2},\ \frac{7 - 3\sqrt{5}}{2} \]
\[ x^2 - 7x + 1 = 0 \]
\[ D = (-7)^2 - 4\cdot1\cdot1 = 49 - 4 = 45 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{45}}{2} \]
\[ x = \frac{7 + 3\sqrt{5}}{2}\quad\text{or}\quad x = \frac{7 - 3\sqrt{5}}{2} \]The parabola y = x² - 7x + 1
| x | y |
|---|---|
| 0.5 | -2.25 |
| 1.5 | -7.25 |
| 2.5 | -10.25 |
| 3.5 | -11.25 |
| 4.5 | -10.25 |
| 5.5 | -7.25 |
| 6.5 | -2.25 |
Questions
What are the solutions of x² - 7x + 1 = 0?
The solutions of x² - 7x + 1 = 0 are x = (7 + 3√5)/2 or x = (7 - 3√5)/2, that is x ≈ 6.854102 and x ≈ 0.14589803.
Does x² - 7x + 1 = 0 factor?
Not over the integers, because the discriminant 45 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² - 7x + 1?
(7/2, -45/4). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 7, product = c/a = 1.