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Solve -2x² + 5x + 12 = 0

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

x² +x += 0
−2x2+5x+12=0⇒x=−32, 4-2x^2 + 5x + 12 = 0 \Rightarrow x = -\frac{3}{2},\ 4
Discriminant D = 121 · x ≈ -1.5, 4

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

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Graph of y = -2x² + 5x + 12-4-2246-40-30-20-101020x = -1.5x = 4vertexy = -2x² + 5x + 12
Graph of y = -2x² + 5x + 12. The curve crosses the x-axis at x = -1.5 and x = 4; 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=0-2x^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⋅(−2)⋅12=25+96=121D = 5^2 - 4\cdot(-2)\cdot12 = 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:

    −(2x+3)(x−4)=0-\left(2x + 3\right)\left(x - 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:

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

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = -3/2 back in: a·x² + b·x + c = -2·(9/4) + 5·(-3/2) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 5x + 12 = 0\ \Rightarrow\ x = -\frac{3}{2},\ 4 \]
\[ -2x^2 + 5x + 12 = 0 \]
\[ D = 5^2 - 4\cdot(-2)\cdot12 = 25 + 96 = 121 \]
\[ -\left(2x + 3\right)\left(x - 4\right) = 0 \]
\[ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x - 4 = 0 \Rightarrow x = 4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{121}}{-4} \]
\[ x = -\frac{3}{2}\quad\text{or}\quad x = 4 \]

The parabola y = -2x² + 5x + 12

xy
-1.75-2.875
-0.757.125
0.2513.125
1.2515.125
2.2513.125
3.257.125
4.25-2.875

Questions

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

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

Does -2x² + 5x + 12 = 0 factor?

Yes: -1(2x+3)(x-4) = 0.

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

(5/4, 121/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = -6.

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