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Solve 2x² - 10x - 12 = 0

The solutions of 2x² - 10x - 12 = 0 are x = 6 or x = -1.

x² +x += 0
2x2−10x−12=0⇒x=6, −12x^2 - 10x - 12 = 0 \Rightarrow x = 6,\ -1
Discriminant D = 196 · x ≈ 6, -1

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

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Graph of y = 2x² - 10x - 12-22468-30-20-101020304050x = 6x = -1vertexy = 2x² - 10x - 12
Graph of y = 2x² - 10x - 12. The curve crosses the x-axis at x = 6 and x = -1; the vertex is at (2.5, -24.5).

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=(−10)2−4⋅2⋅(−12)=100+96=196D = (-10)^2 - 4\cdot2\cdot(-12) = 100 + 96 = 196
  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−6)(x+1)=02\left(x - 6\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−6=0⇒x=6x+1=0⇒x=−1x - 6 = 0 \Rightarrow x = 6\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±1964x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{196}}{4}
  7. 7

    Simplify to get the solutions:

    x=6orx=−1x = 6\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 6 back in: a·x² + b·x + c = 2·(36) + (-10)·(6) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 10x - 12 = 0\ \Rightarrow\ x = 6,\ -1 \]
\[ 2x^2 - 10x - 12 = 0 \]
\[ D = (-10)^2 - 4\cdot2\cdot(-12) = 100 + 96 = 196 \]
\[ 2\left(x - 6\right)\left(x + 1\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{196}}{4} \]
\[ x = 6\quad\text{or}\quad x = -1 \]

The parabola y = 2x² - 10x - 12

xy
-0.5-6.5
0.5-16.5
1.5-22.5
2.5-24.5
3.5-22.5
4.5-16.5
5.5-6.5

Questions

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

The solutions of 2x² - 10x - 12 = 0 are x = 6 or x = -1.

Does 2x² - 10x - 12 = 0 factor?

Yes: 2(x-6)(x+1) = 0.

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

(5/2, -49/2). 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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