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Solve 3x² + 5x - 12 = 0

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

x² +x += 0
3x2+5x−12=0⇒x=43, −33x^2 + 5x - 12 = 0 \Rightarrow x = \frac{4}{3},\ -3
Discriminant D = 169 · x ≈ 1.33333, -3

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

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Graph of y = 3x² + 5x - 12-5-4-3-2-1123-20-101020304050x = 1.3333333x = -3vertexy = 3x² + 5x - 12
Graph of y = 3x² + 5x - 12. The curve crosses the x-axis at x = 1.3333333 and x = -3; 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=52−4⋅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:

    (3x−4)(x+3)=0\left(3x - 4\right)\left(x + 3\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−4=0⇒x=43x+3=0⇒x=−33x - 4 = 0 \Rightarrow x = \frac{4}{3}\qquad x + 3 = 0 \Rightarrow x = -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=43orx=−3x = \frac{4}{3}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 4/3 back in: a·x² + b·x + c = 3·(16/9) + 5·(4/3) + (-12) = 0 ✓

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

The parabola y = 3x² + 5x - 12

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

Questions

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

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

Does 3x² + 5x - 12 = 0 factor?

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