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Solve -x² - 4x + 5 = 0

The solutions of -x² - 4x + 5 = 0 are x = -5 or x = 1.

x² +x += 0
−x2−4x+5=0⇒x=−5, 1-x^2 - 4x + 5 = 0 \Rightarrow x = -5,\ 1
Discriminant D = 36 · x ≈ -5, 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 + 5-6-4-22-20-15-10-5510x = -5x = 1vertexy = -x² - 4x + 5
Graph of y = -x² - 4x + 5. The curve crosses the x-axis at x = -5 and x = 1; the vertex is at (-2, 9).

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 = 5.

    −x2−4x+5=0-x^2 - 4x + 5 = 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)⋅5=16+20=36D = (-4)^2 - 4\cdot(-1)\cdot5 = 16 + 20 = 36
  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+5)(x−1)=0-\left(x + 5\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+5=0⇒x=−5x−1=0⇒x=1x + 5 = 0 \Rightarrow x = -5\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±36−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{36}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−5orx=1x = -5\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -5 back in: a·x² + b·x + c = -1·(25) + (-4)·(-5) + 5 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 4x + 5 = 0\ \Rightarrow\ x = -5,\ 1 \]
\[ -x^2 - 4x + 5 = 0 \]
\[ D = (-4)^2 - 4\cdot(-1)\cdot5 = 16 + 20 = 36 \]
\[ -\left(x + 5\right)\left(x - 1\right) = 0 \]
\[ x + 5 = 0 \Rightarrow x = -5\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{36}}{-2} \]
\[ x = -5\quad\text{or}\quad x = 1 \]

The parabola y = -x² - 4x + 5

xy
-50
-45
-38
-29
-18
05
10

Questions

What are the solutions of -x² - 4x + 5 = 0?

The solutions of -x² - 4x + 5 = 0 are x = -5 or x = 1.

Does -x² - 4x + 5 = 0 factor?

Yes: -1(x+5)(x-1) = 0.

What is the vertex of y = -x² - 4x + 5?

(-2, 9). 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 = -5.

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