Solve 4x² - 7x - 1 = 0
The solutions of 4x² - 7x - 1 = 0 are x = (7 + √65)/8 or x = (7 - √65)/8, that is x ≈ 1.8827822 and x ≈ -0.13278222.
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 = -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 ≈ 1.8827822 and x ≈ -0.13278222.
- 7
Check numerically: for x ≈ 1.8827822 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 4x^2 - 7x - 1 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{65}}{8},\ \frac{7 - \sqrt{65}}{8} \]
\[ 4x^2 - 7x - 1 = 0 \]
\[ D = (-7)^2 - 4\cdot4\cdot(-1) = 49 + 16 = 65 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{65}}{8} \]
\[ x = \frac{7 + \sqrt{65}}{8}\quad\text{or}\quad x = \frac{7 - \sqrt{65}}{8} \]The parabola y = 4x² - 7x - 1
| x | y |
|---|---|
| -2.125 | 31.9375 |
| -1.125 | 11.9375 |
| -0.125 | -0.0625 |
| 0.875 | -4.0625 |
| 1.875 | -0.0625 |
| 2.875 | 11.9375 |
| 3.875 | 31.9375 |
Questions
What are the solutions of 4x² - 7x - 1 = 0?
The solutions of 4x² - 7x - 1 = 0 are x = (7 + √65)/8 or x = (7 - √65)/8, that is x ≈ 1.8827822 and x ≈ -0.13278222.
Does 4x² - 7x - 1 = 0 factor?
Not over the integers, because the discriminant 65 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 4x² - 7x - 1?
(7/8, -65/16). 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 = -1/4.