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Solve 5x² - x - 6 = 0

The solutions of 5x² - x - 6 = 0 are x = 6/5 or x = -1.

x² +x += 0
5x2−x−6=0⇒x=65, −15x^2 - x - 6 = 0 \Rightarrow x = \frac{6}{5},\ -1
Discriminant D = 121 · x ≈ 1.2, -1

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

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Graph of y = 5x² - x - 6-3-2-1123-10102030405060x = 1.2x = -1vertexy = 5x² - x - 6
Graph of y = 5x² - x - 6. The curve crosses the x-axis at x = 1.2 and x = -1; the vertex is at (0.1, -6.05).

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 = -6.

    5x2−x−6=05x^2 - x - 6 = 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⋅(−6)=1+120=121D = (-1)^2 - 4\cdot5\cdot(-6) = 1 + 120 = 121
  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:

    (5x−6)(x+1)=0\left(5x - 6\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:

    5x−6=0⇒x=65x+1=0⇒x=−15x - 6 = 0 \Rightarrow x = \frac{6}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±12110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{121}}{10}
  7. 7

    Simplify to get the solutions:

    x=65orx=−1x = \frac{6}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 6/5 back in: a·x² + b·x + c = 5·(36/25) + (-1)·(6/5) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - x - 6 = 0\ \Rightarrow\ x = \frac{6}{5},\ -1 \]
\[ 5x^2 - x - 6 = 0 \]
\[ D = (-1)^2 - 4\cdot5\cdot(-6) = 1 + 120 = 121 \]
\[ \left(5x - 6\right)\left(x + 1\right) = 0 \]
\[ 5x - 6 = 0 \Rightarrow x = \frac{6}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{121}}{10} \]
\[ x = \frac{6}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - x - 6

xy
-2.938.95
-1.913.95
-0.9-1.05
0.1-6.05
1.1-1.05
2.113.95
3.138.95

Questions

What are the solutions of 5x² - x - 6 = 0?

The solutions of 5x² - x - 6 = 0 are x = 6/5 or x = -1.

Does 5x² - x - 6 = 0 factor?

Yes: (5x-6)(x+1) = 0.

What is the vertex of y = 5x² - x - 6?

(1/10, -121/20). 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 = -6/5.

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