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Solve 5x² - x - 4 = 0

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

x² +x += 0
5x2−x−4=0⇒x=1, −455x^2 - x - 4 = 0 \Rightarrow x = 1,\ -\frac{4}{5}
Discriminant D = 81 · x ≈ 1, -0.8

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

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Graph of y = 5x² - x - 4-3-2-1123-101020304050x = 1x = -0.8vertexy = 5x² - x - 4
Graph of y = 5x² - x - 4. The curve crosses the x-axis at x = 1 and x = -0.8; the vertex is at (0.1, -4.05).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -1, c = -4.

    5x2−x−4=05x^2 - x - 4 = 0
  2. 2

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

    D=(−1)2−4⋅5⋅(−4)=1+80=81D = (-1)^2 - 4\cdot5\cdot(-4) = 1 + 80 = 81
  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−1)(5x+4)=0\left(x - 1\right)\left(5x + 4\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−1=0⇒x=15x+4=0⇒x=−45x - 1 = 0 \Rightarrow x = 1\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±8110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{81}}{10}
  7. 7

    Simplify to get the solutions:

    x=1orx=−45x = 1\quad\text{or}\quad x = -\frac{4}{5}
  8. 8

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

LaTeX source of the formulas on this page
\[ 5x^2 - x - 4 = 0\ \Rightarrow\ x = 1,\ -\frac{4}{5} \]
\[ 5x^2 - x - 4 = 0 \]
\[ D = (-1)^2 - 4\cdot5\cdot(-4) = 1 + 80 = 81 \]
\[ \left(x - 1\right)\left(5x + 4\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 5x + 4 = 0 \Rightarrow x = -\frac{4}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{81}}{10} \]
\[ x = 1\quad\text{or}\quad x = -\frac{4}{5} \]

The parabola y = 5x² - x - 4

xy
-2.940.95
-1.915.95
-0.90.95
0.1-4.05
1.10.95
2.115.95
3.140.95

Questions

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

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

Does 5x² - x - 4 = 0 factor?

Yes: (x-1)(5x+4) = 0.

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

(1/10, -81/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/5, product = c/a = -4/5.

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