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Solve -x² - 4x - 3 = 0

The solutions of -x² - 4x - 3 = 0 are x = -3 or x = -1.

x² +x += 0
−x2−4x−3=0⇒x=−3, −1-x^2 - 4x - 3 = 0 \Rightarrow x = -3,\ -1
Discriminant D = 4 · x ≈ -3, -1

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

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Graph of y = -x² - 4x - 3-5-4-3-2-11-12-10-8-6-4-22x = -3x = -1vertexy = -x² - 4x - 3
Graph of y = -x² - 4x - 3. The curve crosses the x-axis at x = -3 and x = -1; the vertex is at (-2, 1).

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 = -4, c = -3.

    −x2−4x−3=0-x^2 - 4x - 3 = 0
  2. 2

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

    D=(−4)2−4⋅(−1)⋅(−3)=16−12=4D = (-4)^2 - 4\cdot(-1)\cdot(-3) = 16 - 12 = 4
  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+3)(x+1)=0-\left(x + 3\right)\left(x + 1\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+3=0⇒x=−3x+1=0⇒x=−1x + 3 = 0 \Rightarrow x = -3\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±4−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{4}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−3orx=−1x = -3\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -3 back in: a·x² + b·x + c = -1·(9) + (-4)·(-3) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 4x - 3 = 0\ \Rightarrow\ x = -3,\ -1 \]
\[ -x^2 - 4x - 3 = 0 \]
\[ D = (-4)^2 - 4\cdot(-1)\cdot(-3) = 16 - 12 = 4 \]
\[ -\left(x + 3\right)\left(x + 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{4}}{-2} \]
\[ x = -3\quad\text{or}\quad x = -1 \]

The parabola y = -x² - 4x - 3

xy
-5-8
-4-3
-30
-21
-10
0-3
1-8

Questions

What are the solutions of -x² - 4x - 3 = 0?

The solutions of -x² - 4x - 3 = 0 are x = -3 or x = -1.

Does -x² - 4x - 3 = 0 factor?

Yes: -1(x+3)(x+1) = 0.

What is the vertex of y = -x² - 4x - 3?

(-2, 1). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = 3.

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