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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, −1x^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 - 5-2246-10-55101520x = 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=0x^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\cdot1\cdot(-5) = 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±362x = \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
\[ x^2 - 4x - 5 = 0\ \Rightarrow\ x = 5,\ -1 \]
\[ x^2 - 4x - 5 = 0 \]
\[ D = (-4)^2 - 4\cdot1\cdot(-5) = 16 + 20 = 36 \]
\[ \left(x - 5\right)\left(x + 1\right) = 0 \]
\[ x - 5 = 0 \Rightarrow x = 5\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{36}}{2} \]
\[ x = 5\quad\text{or}\quad x = -1 \]

The parabola y = x² - 4x - 5

xy
-10
0-5
1-8
2-9
3-8
4-5
50

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

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

(2, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = -5.

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