Solve 4x² - 9x + 7 = 0
4x² - 9x + 7 = 0 has no real solutions. The two complex solutions are x = (9 + √31 i)/8 and x = (9 - √31 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 = -9, c = 7.
- 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
\[ 4x^2 - 9x + 7 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{31}\, i}{8},\ \frac{9 - \sqrt{31}\, i}{8} \]
\[ 4x^2 - 9x + 7 = 0 \]
\[ D = (-9)^2 - 4\cdot4\cdot7 = 81 - 112 = -31 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{-31}}{8} \]
\[ x = \frac{9 + \sqrt{31}\, i}{8}\quad\text{or}\quad x = \frac{9 - \sqrt{31}\, i}{8} \]The parabola y = 4x² - 9x + 7
| x | y |
|---|---|
| -1.875 | 37.9375 |
| -0.875 | 17.9375 |
| 0.125 | 5.9375 |
| 1.125 | 1.9375 |
| 2.125 | 5.9375 |
| 3.125 | 17.9375 |
| 4.125 | 37.9375 |
Questions
What are the solutions of 4x² - 9x + 7 = 0?
4x² - 9x + 7 = 0 has no real solutions. The two complex solutions are x = (9 + √31 i)/8 and x = (9 - √31 i)/8.
Does 4x² - 9x + 7 = 0 factor?
Not over the integers, because the discriminant -31 is negative. Use the quadratic formula.
What is the vertex of y = 4x² - 9x + 7?
(9/8, 31/16). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 9/4, product = c/a = 7/4.