WordGPT

Solve -x² - 5x - 3 = 0

The solutions of -x² - 5x - 3 = 0 are x = (-5 + √13)/2 or x = (-5 - √13)/2, that is x ≈ -4.3027756 and x ≈ -0.69722436.

x² +x += 0
−x2−5x−3=0⇒x=−5+132, −5−132-x^2 - 5x - 3 = 0 \Rightarrow x = \frac{-5 + \sqrt{13}}{2},\ \frac{-5 - \sqrt{13}}{2}
Discriminant D = 13 · x ≈ -4.30278, -0.697224

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

Explain it step by step in chatMake a short video of it
Graph of y = -x² - 5x - 3-6-5-4-3-2-11-15-10-55x = -4.3027756x = -0.69722436vertexy = -x² - 5x - 3
Graph of y = -x² - 5x - 3. The curve crosses the x-axis at x = -4.3027756 and x = -0.69722436; the vertex is at (-2.5, 3.25).

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 = -5, c = -3.

    −x2−5x−3=0-x^2 - 5x - 3 = 0
  2. 2

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

    D=(−5)2−4⋅(−1)⋅(−3)=25−12=13D = (-5)^2 - 4\cdot(-1)\cdot(-3) = 25 - 12 = 13
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=5±13−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{13}}{-2}
  5. 5

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

    x=−5+132orx=−5−132x = \frac{-5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ -4.3027756 and x ≈ -0.69722436.

  7. 7

    Check numerically: for x ≈ -4.3027756 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 5x - 3 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{13}}{2},\ \frac{-5 - \sqrt{13}}{2} \]
\[ -x^2 - 5x - 3 = 0 \]
\[ D = (-5)^2 - 4\cdot(-1)\cdot(-3) = 25 - 12 = 13 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{13}}{-2} \]
\[ x = \frac{-5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{13}}{2} \]

The parabola y = -x² - 5x - 3

xy
-5.5-5.75
-4.5-0.75
-3.52.25
-2.53.25
-1.52.25
-0.5-0.75
0.5-5.75

Questions

What are the solutions of -x² - 5x - 3 = 0?

The solutions of -x² - 5x - 3 = 0 are x = (-5 + √13)/2 or x = (-5 - √13)/2, that is x ≈ -4.3027756 and x ≈ -0.69722436.

Does -x² - 5x - 3 = 0 factor?

Not over the integers, because the discriminant 13 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 5x - 3?

(-5/2, 13/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 3.

Similar equations

    Solve -x² - 5x - 3 = 0: x = (-5 + √13)/2 or x = (-5 - √13)/2 (steps) | WordGPT