Solve 4x² - 7x + 11 = 0
4x² - 7x + 11 = 0 has no real solutions. The two complex solutions are x = (7 + √127 i)/8 and x = (7 - √127 i)/8.
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 = 4, b = -7, c = 11.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is negative, so there are no real solutions; the two solutions are complex numbers.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
LaTeX source of the formulas on this page
\[ 4x^2 - 7x + 11 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{127}\, i}{8},\ \frac{7 - \sqrt{127}\, i}{8} \]
\[ 4x^2 - 7x + 11 = 0 \]
\[ D = (-7)^2 - 4\cdot4\cdot11 = 49 - 176 = -127 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{-127}}{8} \]
\[ x = \frac{7 + \sqrt{127}\, i}{8}\quad\text{or}\quad x = \frac{7 - \sqrt{127}\, i}{8} \]The parabola y = 4x² - 7x + 11
| x | y |
|---|---|
| -2.125 | 43.9375 |
| -1.125 | 23.9375 |
| -0.125 | 11.9375 |
| 0.875 | 7.9375 |
| 1.875 | 11.9375 |
| 2.875 | 23.9375 |
| 3.875 | 43.9375 |
Questions
What are the solutions of 4x² - 7x + 11 = 0?
4x² - 7x + 11 = 0 has no real solutions. The two complex solutions are x = (7 + √127 i)/8 and x = (7 - √127 i)/8.
Does 4x² - 7x + 11 = 0 factor?
Not over the integers, because the discriminant -127 is negative. Use the quadratic formula.
What is the vertex of y = 4x² - 7x + 11?
(7/8, 127/16). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 7/4, product = c/a = 11/4.