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Solve 3x² + 10x + 7 = 0

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

x² +x += 0
3x2+10x+7=0⇒x=−1, −733x^2 + 10x + 7 = 0 \Rightarrow x = -1,\ -\frac{7}{3}
Discriminant D = 16 · x ≈ -1, -2.33333

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

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Graph of y = 3x² + 10x + 7-4-3-2-11-551015202530x = -1x = -2.3333333vertexy = 3x² + 10x + 7
Graph of y = 3x² + 10x + 7. The curve crosses the x-axis at x = -1 and x = -2.3333333; 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=03x^2 + 10x + 7 = 0
  2. 2

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

    D=102−4⋅3⋅7=100−84=16D = 10^2 - 4\cdot3\cdot7 = 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±166x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{16}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = 3·(1) + 10·(-1) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 10x + 7 = 0\ \Rightarrow\ x = -1,\ -\frac{7}{3} \]
\[ 3x^2 + 10x + 7 = 0 \]
\[ D = 10^2 - 4\cdot3\cdot7 = 100 - 84 = 16 \]
\[ \left(x + 1\right)\left(3x + 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 7 = 0 \Rightarrow x = -\frac{7}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{16}}{6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{7}{3} \]

The parabola y = 3x² + 10x + 7

xy
-4.666725.667267
-3.666710.667067
-2.66671.6668667
-1.6667-1.3333333
-0.66671.6664667
0.333310.666267
1.333325.666067

Questions

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

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

Does 3x² + 10x + 7 = 0 factor?

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

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

(-5/3, -4/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -10/3, product = c/a = 7/3.

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