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Solve 3x² - 9x - 12 = 0

The solutions of 3x² - 9x - 12 = 0 are x = 4 or x = -1.

x² +x += 0
3x2−9x−12=0⇒x=4, −13x^2 - 9x - 12 = 0 \Rightarrow x = 4,\ -1
Discriminant D = 225 · x ≈ 4, -1

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

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Graph of y = 3x² - 9x - 12-3-2-1123456-20204060x = 4x = -1vertexy = 3x² - 9x - 12
Graph of y = 3x² - 9x - 12. The curve crosses the x-axis at x = 4 and x = -1; the vertex is at (1.5, -18.75).

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 = -9, c = -12.

    3x2−9x−12=03x^2 - 9x - 12 = 0
  2. 2

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

    D=(−9)2−4⋅3⋅(−12)=81+144=225D = (-9)^2 - 4\cdot3\cdot(-12) = 81 + 144 = 225
  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:

    3(x−4)(x+1)=03\left(x - 4\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:

    x−4=0⇒x=4x+1=0⇒x=−1x - 4 = 0 \Rightarrow x = 4\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±2256x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{6}
  7. 7

    Simplify to get the solutions:

    x=4orx=−1x = 4\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 3·(16) + (-9)·(4) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 9x - 12 = 0\ \Rightarrow\ x = 4,\ -1 \]
\[ 3x^2 - 9x - 12 = 0 \]
\[ D = (-9)^2 - 4\cdot3\cdot(-12) = 81 + 144 = 225 \]
\[ 3\left(x - 4\right)\left(x + 1\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{225}}{6} \]
\[ x = 4\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 9x - 12

xy
-1.58.25
-0.5-6.75
0.5-15.75
1.5-18.75
2.5-15.75
3.5-6.75
4.58.25

Questions

What are the solutions of 3x² - 9x - 12 = 0?

The solutions of 3x² - 9x - 12 = 0 are x = 4 or x = -1.

Does 3x² - 9x - 12 = 0 factor?

Yes: 3(x-4)(x+1) = 0.

What is the vertex of y = 3x² - 9x - 12?

(3/2, -75/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -4.

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