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Solve 3x² - 7x - 10 = 0

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

x² +x += 0
3x2−7x−10=0⇒x=103, −13x^2 - 7x - 10 = 0 \Rightarrow x = \frac{10}{3},\ -1
Discriminant D = 169 · x ≈ 3.33333, -1

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

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Graph of y = 3x² - 7x - 10-3-2-112345-20-101020304050x = 3.3333333x = -1vertexy = 3x² - 7x - 10
Graph of y = 3x² - 7x - 10. The curve crosses the x-axis at x = 3.3333333 and x = -1; the vertex is at (1.1666667, -14.083333).

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 = -7, c = -10.

    3x2−7x−10=03x^2 - 7x - 10 = 0
  2. 2

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

    D=(−7)2−4⋅3⋅(−10)=49+120=169D = (-7)^2 - 4\cdot3\cdot(-10) = 49 + 120 = 169
  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−10)(x+1)=0\left(3x - 10\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−10=0⇒x=103x+1=0⇒x=−13x - 10 = 0 \Rightarrow x = \frac{10}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±1696x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{169}}{6}
  7. 7

    Simplify to get the solutions:

    x=103orx=−1x = \frac{10}{3}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 10/3 back in: a·x² + b·x + c = 3·(100/9) + (-7)·(10/3) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 7x - 10 = 0\ \Rightarrow\ x = \frac{10}{3},\ -1 \]
\[ 3x^2 - 7x - 10 = 0 \]
\[ D = (-7)^2 - 4\cdot3\cdot(-10) = 49 + 120 = 169 \]
\[ \left(3x - 10\right)\left(x + 1\right) = 0 \]
\[ 3x - 10 = 0 \Rightarrow x = \frac{10}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{169}}{6} \]
\[ x = \frac{10}{3}\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 7x - 10

xy
-1.833312.916067
-0.8333-2.0837333
0.1667-11.083533
1.1667-14.083333
2.1667-11.083133
3.1667-2.0829333
4.166712.917267

Questions

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

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

Does 3x² - 7x - 10 = 0 factor?

Yes: (3x-10)(x+1) = 0.

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

(7/6, -169/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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