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Solve 3x² - 5x - 12 = 0

The solutions of 3x² - 5x - 12 = 0 are x = 3 or x = -4/3.

x² +x += 0
3x2−5x−12=0⇒x=3, −433x^2 - 5x - 12 = 0 \Rightarrow x = 3,\ -\frac{4}{3}
Discriminant D = 169 · x ≈ 3, -1.33333

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

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Graph of y = 3x² - 5x - 12-3-2-112345-20-101020304050x = 3x = -1.3333333vertexy = 3x² - 5x - 12
Graph of y = 3x² - 5x - 12. The curve crosses the x-axis at x = 3 and x = -1.3333333; the vertex is at (0.83333333, -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 = -5, c = -12.

    3x2−5x−12=03x^2 - 5x - 12 = 0
  2. 2

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

    D=(−5)2−4⋅3⋅(−12)=25+144=169D = (-5)^2 - 4\cdot3\cdot(-12) = 25 + 144 = 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:

    (x−3)(3x+4)=0\left(x - 3\right)\left(3x + 4\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+4=0⇒x=−43x - 3 = 0 \Rightarrow x = 3\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1696x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{169}}{6}
  7. 7

    Simplify to get the solutions:

    x=3orx=−43x = 3\quad\text{or}\quad x = -\frac{4}{3}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 3·(9) + (-5)·(3) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 5x - 12 = 0\ \Rightarrow\ x = 3,\ -\frac{4}{3} \]
\[ 3x^2 - 5x - 12 = 0 \]
\[ D = (-5)^2 - 4\cdot3\cdot(-12) = 25 + 144 = 169 \]
\[ \left(x - 3\right)\left(3x + 4\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{169}}{6} \]
\[ x = 3\quad\text{or}\quad x = -\frac{4}{3} \]

The parabola y = 3x² - 5x - 12

xy
-2.166712.917267
-1.1667-2.0829333
-0.1667-11.083133
0.8333-14.083333
1.8333-11.083533
2.8333-2.0837333
3.833312.916067

Questions

What are the solutions of 3x² - 5x - 12 = 0?

The solutions of 3x² - 5x - 12 = 0 are x = 3 or x = -4/3.

Does 3x² - 5x - 12 = 0 factor?

Yes: (x-3)(3x+4) = 0.

What is the vertex of y = 3x² - 5x - 12?

(5/6, -169/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = -4.

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