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Solve 5x² - 6x + 1 = 0

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

x² +x += 0
5x2−6x+1=0⇒x=1, 155x^2 - 6x + 1 = 0 \Rightarrow x = 1,\ \frac{1}{5}
Discriminant D = 16 · x ≈ 1, 0.2

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

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Graph of y = 5x² - 6x + 1-2-112310203040x = 1x = 0.2vertexy = 5x² - 6x + 1
Graph of y = 5x² - 6x + 1. The curve crosses the x-axis at x = 1 and x = 0.2; the vertex is at (0.6, -0.8).

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 = -6, c = 1.

    5x2−6x+1=05x^2 - 6x + 1 = 0
  2. 2

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

    D=(−6)2−4⋅5⋅1=36−20=16D = (-6)^2 - 4\cdot5\cdot1 = 36 - 20 = 16
  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−1)=0\left(x - 1\right)\left(5x - 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−1=0⇒x=15x−1=0⇒x=15x - 1 = 0 \Rightarrow x = 1\qquad 5x - 1 = 0 \Rightarrow x = \frac{1}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±1610x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{16}}{10}
  7. 7

    Simplify to get the solutions:

    x=1orx=15x = 1\quad\text{or}\quad x = \frac{1}{5}
  8. 8

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

LaTeX source of the formulas on this page
\[ 5x^2 - 6x + 1 = 0\ \Rightarrow\ x = 1,\ \frac{1}{5} \]
\[ 5x^2 - 6x + 1 = 0 \]
\[ D = (-6)^2 - 4\cdot5\cdot1 = 36 - 20 = 16 \]
\[ \left(x - 1\right)\left(5x - 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 5x - 1 = 0 \Rightarrow x = \frac{1}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{16}}{10} \]
\[ x = 1\quad\text{or}\quad x = \frac{1}{5} \]

The parabola y = 5x² - 6x + 1

xy
-2.444.2
-1.419.2
-0.44.2
0.6-0.8
1.64.2
2.619.2
3.644.2

Questions

What are the solutions of 5x² - 6x + 1 = 0?

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

Does 5x² - 6x + 1 = 0 factor?

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

What is the vertex of y = 5x² - 6x + 1?

(3/5, -4/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6/5, product = c/a = 1/5.

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