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Solve 2x² - 5x - 12 = 0

The solutions of 2x² - 5x - 12 = 0 are x = 4 or x = -3/2.

x² +x += 0
2x2−5x−12=0⇒x=4, −322x^2 - 5x - 12 = 0 \Rightarrow x = 4,\ -\frac{3}{2}
Discriminant D = 121 · x ≈ 4, -1.5

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

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Graph of y = 2x² - 5x - 12-4-2246-20-1010203040x = 4x = -1.5vertexy = 2x² - 5x - 12
Graph of y = 2x² - 5x - 12. The curve crosses the x-axis at x = 4 and x = -1.5; the vertex is at (1.25, -15.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -5, c = -12.

    2x2−5x−12=02x^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⋅2⋅(−12)=25+96=121D = (-5)^2 - 4\cdot2\cdot(-12) = 25 + 96 = 121
  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−4)(2x+3)=0\left(x - 4\right)\left(2x + 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:

    x−4=0⇒x=42x+3=0⇒x=−32x - 4 = 0 \Rightarrow x = 4\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±1214x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{4}
  7. 7

    Simplify to get the solutions:

    x=4orx=−32x = 4\quad\text{or}\quad x = -\frac{3}{2}
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 2·(16) + (-5)·(4) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 5x - 12 = 0\ \Rightarrow\ x = 4,\ -\frac{3}{2} \]
\[ 2x^2 - 5x - 12 = 0 \]
\[ D = (-5)^2 - 4\cdot2\cdot(-12) = 25 + 96 = 121 \]
\[ \left(x - 4\right)\left(2x + 3\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{4} \]
\[ x = 4\quad\text{or}\quad x = -\frac{3}{2} \]

The parabola y = 2x² - 5x - 12

xy
-1.752.875
-0.75-7.125
0.25-13.125
1.25-15.125
2.25-13.125
3.25-7.125
4.252.875

Questions

What are the solutions of 2x² - 5x - 12 = 0?

The solutions of 2x² - 5x - 12 = 0 are x = 4 or x = -3/2.

Does 2x² - 5x - 12 = 0 factor?

Yes: (x-4)(2x+3) = 0.

What is the vertex of y = 2x² - 5x - 12?

(5/4, -121/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = -6.

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