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Solve -3x² - 10x - 3 = 0

The solutions of -3x² - 10x - 3 = 0 are x = -3 or x = -1/3.

x² +x += 0
−3x2−10x−3=0⇒x=−3, −13-3x^2 - 10x - 3 = 0 \Rightarrow x = -3,\ -\frac{1}{3}
Discriminant D = 64 · x ≈ -3, -0.333333

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -3x² - 10x - 3-5-4-3-2-112-40-30-20-1010x = -3x = -0.33333333vertexy = -3x² - 10x - 3
Graph of y = -3x² - 10x - 3. The curve crosses the x-axis at x = -3 and x = -0.33333333; the vertex is at (-1.6666667, 5.3333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = -10, c = -3.

    −3x2−10x−3=0-3x^2 - 10x - 3 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−10)2−4⋅(−3)⋅(−3)=100−36=64D = (-10)^2 - 4\cdot(-3)\cdot(-3) = 100 - 36 = 64
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x+3)(3x+1)=0-\left(x + 3\right)\left(3x + 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+3=0⇒x=−33x+1=0⇒x=−13x + 3 = 0 \Rightarrow x = -3\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±64−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{64}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−3orx=−13x = -3\quad\text{or}\quad x = -\frac{1}{3}
  8. 8

    Check by putting x = -3 back in: a·x² + b·x + c = -3·(9) + (-10)·(-3) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 10x - 3 = 0\ \Rightarrow\ x = -3,\ -\frac{1}{3} \]
\[ -3x^2 - 10x - 3 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot(-3) = 100 - 36 = 64 \]
\[ -\left(x + 3\right)\left(3x + 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{64}}{-6} \]
\[ x = -3\quad\text{or}\quad x = -\frac{1}{3} \]

The parabola y = -3x² - 10x - 3

xy
-4.6667-21.667267
-3.6667-6.6670667
-2.66672.3331333
-1.66675.3333333
-0.66672.3335333
0.3333-6.6662667
1.3333-21.666067

Questions

What are the solutions of -3x² - 10x - 3 = 0?

The solutions of -3x² - 10x - 3 = 0 are x = -3 or x = -1/3.

Does -3x² - 10x - 3 = 0 factor?

Yes: -1(x+3)(3x+1) = 0.

What is the vertex of y = -3x² - 10x - 3?

(-5/3, 16/3). 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.

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