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Solve x² - 8x - 12 = 0

The solutions of x² - 8x - 12 = 0 are x = 4 + 2√7 or x = 4 - 2√7, that is x ≈ 9.2915026 and x ≈ -1.2915026.

x² +x += 0
x2−8x−12=0⇒x=4+27, 4−27x^2 - 8x - 12 = 0 \Rightarrow x = 4 + 2\sqrt{7},\ 4 - 2\sqrt{7}
Discriminant D = 112 · x ≈ 9.2915, -1.2915

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

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Graph of y = x² - 8x - 12-2246810-30-20-10102030x = 9.2915026x = -1.2915026vertexy = x² - 8x - 12
Graph of y = x² - 8x - 12. The curve crosses the x-axis at x = 9.2915026 and x = -1.2915026; the vertex is at (4, -28).

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=112D = (-8)^2 - 4\cdot1\cdot(-12) = 64 + 48 = 112
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=8±1122x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{112}}{2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=4+27orx=4−27x = 4 + 2\sqrt{7}\quad\text{or}\quad x = 4 - 2\sqrt{7}
  6. 6

    As decimals: x ≈ 9.2915026 and x ≈ -1.2915026.

  7. 7

    Check numerically: for x ≈ 9.2915026 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 12 = 0\ \Rightarrow\ x = 4 + 2\sqrt{7},\ 4 - 2\sqrt{7} \]
\[ x^2 - 8x - 12 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-12) = 64 + 48 = 112 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{112}}{2} \]
\[ x = 4 + 2\sqrt{7}\quad\text{or}\quad x = 4 - 2\sqrt{7} \]

The parabola y = x² - 8x - 12

xy
1-19
2-24
3-27
4-28
5-27
6-24
7-19

Questions

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

The solutions of x² - 8x - 12 = 0 are x = 4 + 2√7 or x = 4 - 2√7, that is x ≈ 9.2915026 and x ≈ -1.2915026.

Does x² - 8x - 12 = 0 factor?

Not over the integers, because the discriminant 112 is not a perfect square. Use the quadratic formula.

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

(4, -28). 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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    Solve x² - 8x - 12 = 0: x = 4 + 2√7 or x = 4 - 2√7 (steps) | WordGPT