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Solve 2x² + 10x + 12 = 0

The solutions of 2x² + 10x + 12 = 0 are x = -2 or x = -3.

x² +x += 0
2x2+10x+12=0⇒x=−2, −32x^2 + 10x + 12 = 0 \Rightarrow x = -2,\ -3
Discriminant D = 4 · x ≈ -2, -3

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

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Graph of y = 2x² + 10x + 12-5-4-3-2-151015x = -2x = -3vertexy = 2x² + 10x + 12
Graph of y = 2x² + 10x + 12. The curve crosses the x-axis at x = -2 and x = -3; the vertex is at (-2.5, -0.5).

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 = 10, c = 12.

    2x2+10x+12=02x^2 + 10x + 12 = 0
  2. 2

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

    D=102−4⋅2⋅12=100−96=4D = 10^2 - 4\cdot2\cdot12 = 100 - 96 = 4
  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:

    2(x+2)(x+3)=02\left(x + 2\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:

    x+2=0⇒x=−2x+3=0⇒x=−3x + 2 = 0 \Rightarrow x = -2\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±44x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -2 back in: a·x² + b·x + c = 2·(4) + 10·(-2) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 10x + 12 = 0\ \Rightarrow\ x = -2,\ -3 \]
\[ 2x^2 + 10x + 12 = 0 \]
\[ D = 10^2 - 4\cdot2\cdot12 = 100 - 96 = 4 \]
\[ 2\left(x + 2\right)\left(x + 3\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{4} \]
\[ x = -2\quad\text{or}\quad x = -3 \]

The parabola y = 2x² + 10x + 12

xy
-5.517.5
-4.57.5
-3.51.5
-2.5-0.5
-1.51.5
-0.57.5
0.517.5

Questions

What are the solutions of 2x² + 10x + 12 = 0?

The solutions of 2x² + 10x + 12 = 0 are x = -2 or x = -3.

Does 2x² + 10x + 12 = 0 factor?

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

What is the vertex of y = 2x² + 10x + 12?

(-5/2, -1/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 6.

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