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Solve -3x² - 10x - 7 = 0

The solutions of -3x² - 10x - 7 = 0 are x = -7/3 or x = -1.

x² +x += 0
−3x2−10x−7=0⇒x=−73, −1-3x^2 - 10x - 7 = 0 \Rightarrow x = -\frac{7}{3},\ -1
Discriminant D = 16 · x ≈ -2.33333, -1

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

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Graph of y = -3x² - 10x - 7-4-3-2-11-30-25-20-15-10-55x = -2.3333333x = -1vertexy = -3x² - 10x - 7
Graph of y = -3x² - 10x - 7. The curve crosses the x-axis at x = -2.3333333 and x = -1; the vertex is at (-1.6666667, 1.3333333).

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 = -7.

    −3x2−10x−7=0-3x^2 - 10x - 7 = 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)⋅(−7)=100−84=16D = (-10)^2 - 4\cdot(-3)\cdot(-7) = 100 - 84 = 16
  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:

    −(3x+7)(x+1)=0-\left(3x + 7\right)\left(x + 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:

    3x+7=0⇒x=−73x+1=0⇒x=−13x + 7 = 0 \Rightarrow x = -\frac{7}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±16−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{16}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−73orx=−1x = -\frac{7}{3}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -7/3 back in: a·x² + b·x + c = -3·(49/9) + (-10)·(-7/3) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 10x - 7 = 0\ \Rightarrow\ x = -\frac{7}{3},\ -1 \]
\[ -3x^2 - 10x - 7 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot(-7) = 100 - 84 = 16 \]
\[ -\left(3x + 7\right)\left(x + 1\right) = 0 \]
\[ 3x + 7 = 0 \Rightarrow x = -\frac{7}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{16}}{-6} \]
\[ x = -\frac{7}{3}\quad\text{or}\quad x = -1 \]

The parabola y = -3x² - 10x - 7

xy
-4.6667-25.667267
-3.6667-10.667067
-2.6667-1.6668667
-1.66671.3333333
-0.6667-1.6664667
0.3333-10.666267
1.3333-25.666067

Questions

What are the solutions of -3x² - 10x - 7 = 0?

The solutions of -3x² - 10x - 7 = 0 are x = -7/3 or x = -1.

Does -3x² - 10x - 7 = 0 factor?

Yes: -1(3x+7)(x+1) = 0.

What is the vertex of y = -3x² - 10x - 7?

(-5/3, 4/3). 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 = 7/3.

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