Solve -3x² - 10x - 1 = 0
The solutions of -3x² - 10x - 1 = 0 are x = (-5 + √22)/3 or x = (-5 - √22)/3, that is x ≈ -3.2301386 and x ≈ -0.10319475.
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 = -3, b = -10, 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 ≈ -3.2301386 and x ≈ -0.10319475.
- 7
Check numerically: for x ≈ -3.2301386 the left side is -3.5527137e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -3x^2 - 10x - 1 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{22}}{3},\ \frac{-5 - \sqrt{22}}{3} \]
\[ -3x^2 - 10x - 1 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot(-1) = 100 - 12 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{88}}{-6} \]
\[ x = \frac{-5 + \sqrt{22}}{3}\quad\text{or}\quad x = \frac{-5 - \sqrt{22}}{3} \]The parabola y = -3x² - 10x - 1
| x | y |
|---|---|
| -4.6667 | -19.667267 |
| -3.6667 | -4.6670667 |
| -2.6667 | 4.3331333 |
| -1.6667 | 7.3333333 |
| -0.6667 | 4.3335333 |
| 0.3333 | -4.6662667 |
| 1.3333 | -19.666067 |
Questions
What are the solutions of -3x² - 10x - 1 = 0?
The solutions of -3x² - 10x - 1 = 0 are x = (-5 + √22)/3 or x = (-5 - √22)/3, that is x ≈ -3.2301386 and x ≈ -0.10319475.
Does -3x² - 10x - 1 = 0 factor?
Not over the integers, because the discriminant 88 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -3x² - 10x - 1?
(-5/3, 22/3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -10/3, product = c/a = 1/3.