Solve 4x² + 4x - 11 = 0
The solutions of 4x² + 4x - 11 = 0 are x = (-1 + 2√3)/2 or x = (-1 - 2√3)/2, that is x ≈ 1.2320508 and x ≈ -2.2320508.
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 = 4, c = -11.
- 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 ≈ 1.2320508 and x ≈ -2.2320508.
- 7
Check numerically: for x ≈ 1.2320508 the left side is -1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 4x^2 + 4x - 11 = 0\ \Rightarrow\ x = \frac{-1 + 2\sqrt{3}}{2},\ \frac{-1 - 2\sqrt{3}}{2} \]
\[ 4x^2 + 4x - 11 = 0 \]
\[ D = 4^2 - 4\cdot4\cdot(-11) = 16 + 176 = 192 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{192}}{8} \]
\[ x = \frac{-1 + 2\sqrt{3}}{2}\quad\text{or}\quad x = \frac{-1 - 2\sqrt{3}}{2} \]The parabola y = 4x² + 4x - 11
| x | y |
|---|---|
| -3.5 | 24 |
| -2.5 | 4 |
| -1.5 | -8 |
| -0.5 | -12 |
| 0.5 | -8 |
| 1.5 | 4 |
| 2.5 | 24 |
Questions
What are the solutions of 4x² + 4x - 11 = 0?
The solutions of 4x² + 4x - 11 = 0 are x = (-1 + 2√3)/2 or x = (-1 - 2√3)/2, that is x ≈ 1.2320508 and x ≈ -2.2320508.
Does 4x² + 4x - 11 = 0 factor?
Not over the integers, because the discriminant 192 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 4x² + 4x - 11?
(-1/2, -12). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -1, product = c/a = -11/4.