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Solve x² - 8x + 12 = 0

The solutions of x² - 8x + 12 = 0 are x = 6 or x = 2.

x² +x += 0
x2−8x+12=0⇒x=6, 2x^2 - 8x + 12 = 0 \Rightarrow x = 6,\ 2
Discriminant D = 16 · x ≈ 6, 2

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

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Graph of y = x² - 8x + 1212345678-551015x = 6x = 2vertexy = x² - 8x + 12
Graph of y = x² - 8x + 12. The curve crosses the x-axis at x = 6 and x = 2; the vertex is at (4, -4).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -8, c = 12.

    x2−8x+12=0x^2 - 8x + 12 = 0
  2. 2

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

    D=(−8)2−4⋅1⋅12=64−48=16D = (-8)^2 - 4\cdot1\cdot12 = 64 - 48 = 16
  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−6)(x−2)=0\left(x - 6\right)\left(x - 2\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−2=0⇒x=2x - 6 = 0 \Rightarrow x = 6\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=8±162x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{16}}{2}
  7. 7

    Simplify to get the solutions:

    x=6orx=2x = 6\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = 6 back in: a·x² + b·x + c = 1·(36) + (-8)·(6) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x + 12 = 0\ \Rightarrow\ x = 6,\ 2 \]
\[ x^2 - 8x + 12 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot12 = 64 - 48 = 16 \]
\[ \left(x - 6\right)\left(x - 2\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{16}}{2} \]
\[ x = 6\quad\text{or}\quad x = 2 \]

The parabola y = x² - 8x + 12

xy
15
20
3-3
4-4
5-3
60
75

Questions

What are the solutions of x² - 8x + 12 = 0?

The solutions of x² - 8x + 12 = 0 are x = 6 or x = 2.

Does x² - 8x + 12 = 0 factor?

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

What is the vertex of y = x² - 8x + 12?

(4, -4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = 12.

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