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Solve 5x² - 9x - 2 = 0

The solutions of 5x² - 9x - 2 = 0 are x = 2 or x = -1/5.

x² +x += 0
5x2−9x−2=0⇒x=2, −155x^2 - 9x - 2 = 0 \Rightarrow x = 2,\ -\frac{1}{5}
Discriminant D = 121 · x ≈ 2, -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 - 9x - 2-2-11234-10102030405060x = 2x = -0.2vertexy = 5x² - 9x - 2
Graph of y = 5x² - 9x - 2. The curve crosses the x-axis at x = 2 and x = -0.2; the vertex is at (0.9, -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 = -9, c = -2.

    5x2−9x−2=05x^2 - 9x - 2 = 0
  2. 2

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

    D=(−9)2−4⋅5⋅(−2)=81+40=121D = (-9)^2 - 4\cdot5\cdot(-2) = 81 + 40 = 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:

    (x−2)(5x+1)=0\left(x - 2\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−2=0⇒x=25x+1=0⇒x=−15x - 2 = 0 \Rightarrow x = 2\qquad 5x + 1 = 0 \Rightarrow x = -\frac{1}{5}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±12110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{121}}{10}
  7. 7

    Simplify to get the solutions:

    x=2orx=−15x = 2\quad\text{or}\quad x = -\frac{1}{5}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 5·(4) + (-9)·(2) + (-2) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 9x - 2 = 0\ \Rightarrow\ x = 2,\ -\frac{1}{5} \]
\[ 5x^2 - 9x - 2 = 0 \]
\[ D = (-9)^2 - 4\cdot5\cdot(-2) = 81 + 40 = 121 \]
\[ \left(x - 2\right)\left(5x + 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 5x + 1 = 0 \Rightarrow x = -\frac{1}{5} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{121}}{10} \]
\[ x = 2\quad\text{or}\quad x = -\frac{1}{5} \]

The parabola y = 5x² - 9x - 2

xy
-2.138.95
-1.113.95
-0.1-1.05
0.9-6.05
1.9-1.05
2.913.95
3.938.95

Questions

What are the solutions of 5x² - 9x - 2 = 0?

The solutions of 5x² - 9x - 2 = 0 are x = 2 or x = -1/5.

Does 5x² - 9x - 2 = 0 factor?

Yes: (x-2)(5x+1) = 0.

What is the vertex of y = 5x² - 9x - 2?

(9/10, -121/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9/5, product = c/a = -2/5.

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