Solve x² + 12x + 5 = 0
The solutions of x² + 12x + 5 = 0 are x = -6 + √31 or x = -6 - √31, that is x ≈ -0.43223564 and x ≈ -11.567764.
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 = 1, b = 12, c = 5.
- 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 ≈ -0.43223564 and x ≈ -11.567764.
- 7
Check numerically: for x ≈ -0.43223564 the left side is -4.4408921e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ x^2 + 12x + 5 = 0\ \Rightarrow\ x = -6 + \sqrt{31},\ -6 - \sqrt{31} \]
\[ x^2 + 12x + 5 = 0 \]
\[ D = 12^2 - 4\cdot1\cdot5 = 144 - 20 = 124 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{124}}{2} \]
\[ x = -6 + \sqrt{31}\quad\text{or}\quad x = -6 - \sqrt{31} \]The parabola y = x² + 12x + 5
| x | y |
|---|---|
| -9 | -22 |
| -8 | -27 |
| -7 | -30 |
| -6 | -31 |
| -5 | -30 |
| -4 | -27 |
| -3 | -22 |
Questions
What are the solutions of x² + 12x + 5 = 0?
The solutions of x² + 12x + 5 = 0 are x = -6 + √31 or x = -6 - √31, that is x ≈ -0.43223564 and x ≈ -11.567764.
Does x² + 12x + 5 = 0 factor?
Not over the integers, because the discriminant 124 is not a perfect square. Use the quadratic formula.
What is the vertex of y = x² + 12x + 5?
(-6, -31). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -12, product = c/a = 5.