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Solve -x² - 12x + 4 = 0

The solutions of -x² - 12x + 4 = 0 are x = -6 + 2√10 or x = -6 - 2√10, that is x ≈ -12.324555 and x ≈ 0.32455532.

x² +x += 0
−x2−12x+4=0⇒x=−6+210, −6−210-x^2 - 12x + 4 = 0 \Rightarrow x = -6 + 2\sqrt{10},\ -6 - 2\sqrt{10}
Discriminant D = 160 · x ≈ -12.3246, 0.324555

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

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Graph of y = -x² - 12x + 4-14-12-10-8-6-4-22-40-202040x = -12.324555x = 0.32455532vertexy = -x² - 12x + 4
Graph of y = -x² - 12x + 4. The curve crosses the x-axis at x = -12.324555 and x = 0.32455532; the vertex is at (-6, 40).

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 = -12, c = 4.

    −x2−12x+4=0-x^2 - 12x + 4 = 0
  2. 2

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

    D=(−12)2−4⋅(−1)⋅4=144+16=160D = (-12)^2 - 4\cdot(-1)\cdot4 = 144 + 16 = 160
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±160−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{160}}{-2}
  5. 5

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

    x=−6+210orx=−6−210x = -6 + 2\sqrt{10}\quad\text{or}\quad x = -6 - 2\sqrt{10}
  6. 6

    As decimals: x ≈ -12.324555 and x ≈ 0.32455532.

  7. 7

    Check numerically: for x ≈ -12.324555 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 12x + 4 = 0\ \Rightarrow\ x = -6 + 2\sqrt{10},\ -6 - 2\sqrt{10} \]
\[ -x^2 - 12x + 4 = 0 \]
\[ D = (-12)^2 - 4\cdot(-1)\cdot4 = 144 + 16 = 160 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{160}}{-2} \]
\[ x = -6 + 2\sqrt{10}\quad\text{or}\quad x = -6 - 2\sqrt{10} \]

The parabola y = -x² - 12x + 4

xy
-931
-836
-739
-640
-539
-436
-331

Questions

What are the solutions of -x² - 12x + 4 = 0?

The solutions of -x² - 12x + 4 = 0 are x = -6 + 2√10 or x = -6 - 2√10, that is x ≈ -12.324555 and x ≈ 0.32455532.

Does -x² - 12x + 4 = 0 factor?

Not over the integers, because the discriminant 160 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 12x + 4?

(-6, 40). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -12, product = c/a = -4.

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    Solve -x² - 12x + 4 = 0: x = -6 + 2√10 or x = -6 - 2√10 (steps) | WordGPT