Solve 4x² + 3x + 5 = 0
4x² + 3x + 5 = 0 has no real solutions. The two complex solutions are x = (-3 + √71 i)/8 and x = (-3 - √71 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 = 3, c = 5.
- 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
\[ 4x^2 + 3x + 5 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{71}\, i}{8},\ \frac{-3 - \sqrt{71}\, i}{8} \]
\[ 4x^2 + 3x + 5 = 0 \]
\[ D = 3^2 - 4\cdot4\cdot5 = 9 - 80 = -71 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{-71}}{8} \]
\[ x = \frac{-3 + \sqrt{71}\, i}{8}\quad\text{or}\quad x = \frac{-3 - \sqrt{71}\, i}{8} \]The parabola y = 4x² + 3x + 5
| x | y |
|---|---|
| -3.375 | 40.4375 |
| -2.375 | 20.4375 |
| -1.375 | 8.4375 |
| -0.375 | 4.4375 |
| 0.625 | 8.4375 |
| 1.625 | 20.4375 |
| 2.625 | 40.4375 |
Questions
What are the solutions of 4x² + 3x + 5 = 0?
4x² + 3x + 5 = 0 has no real solutions. The two complex solutions are x = (-3 + √71 i)/8 and x = (-3 - √71 i)/8.
Does 4x² + 3x + 5 = 0 factor?
Not over the integers, because the discriminant -71 is negative. Use the quadratic formula.
What is the vertex of y = 4x² + 3x + 5?
(-3/8, 71/16). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -3/4, product = c/a = 5/4.