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Solve -x² - 8x - 3 = 0

The solutions of -x² - 8x - 3 = 0 are x = -4 + √13 or x = -4 - √13, that is x ≈ -7.6055513 and x ≈ -0.39444872.

x² +x += 0
−x2−8x−3=0⇒x=−4+13, −4−13-x^2 - 8x - 3 = 0 \Rightarrow x = -4 + \sqrt{13},\ -4 - \sqrt{13}
Discriminant D = 52 · x ≈ -7.60555, -0.394449

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

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Graph of y = -x² - 8x - 3-10-8-6-4-22-25-20-15-10-551015x = -7.6055513x = -0.39444872vertexy = -x² - 8x - 3
Graph of y = -x² - 8x - 3. The curve crosses the x-axis at x = -7.6055513 and x = -0.39444872; the vertex is at (-4, 13).

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 = -3.

    −x2−8x−3=0-x^2 - 8x - 3 = 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)⋅(−3)=64−12=52D = (-8)^2 - 4\cdot(-1)\cdot(-3) = 64 - 12 = 52
  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±52−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{52}}{-2}
  5. 5

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

    x=−4+13orx=−4−13x = -4 + \sqrt{13}\quad\text{or}\quad x = -4 - \sqrt{13}
  6. 6

    As decimals: x ≈ -7.6055513 and x ≈ -0.39444872.

  7. 7

    Check numerically: for x ≈ -7.6055513 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 8x - 3 = 0\ \Rightarrow\ x = -4 + \sqrt{13},\ -4 - \sqrt{13} \]
\[ -x^2 - 8x - 3 = 0 \]
\[ D = (-8)^2 - 4\cdot(-1)\cdot(-3) = 64 - 12 = 52 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{52}}{-2} \]
\[ x = -4 + \sqrt{13}\quad\text{or}\quad x = -4 - \sqrt{13} \]

The parabola y = -x² - 8x - 3

xy
-74
-69
-512
-413
-312
-29
-14

Questions

What are the solutions of -x² - 8x - 3 = 0?

The solutions of -x² - 8x - 3 = 0 are x = -4 + √13 or x = -4 - √13, that is x ≈ -7.6055513 and x ≈ -0.39444872.

Does -x² - 8x - 3 = 0 factor?

Not over the integers, because the discriminant 52 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 8x - 3?

(-4, 13). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -8, product = c/a = 3.

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    Solve -x² - 8x - 3 = 0: x = -4 + √13 or x = -4 - √13 (steps) | WordGPT