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Solve 2x² - 5x - 6 = 0

The solutions of 2x² - 5x - 6 = 0 are x = (5 + √73)/4 or x = (5 - √73)/4, that is x ≈ 3.3860009 and x ≈ -0.88600094.

x² +x += 0
2x2−5x−6=0⇒x=5+734, 5−7342x^2 - 5x - 6 = 0 \Rightarrow x = \frac{5 + \sqrt{73}}{4},\ \frac{5 - \sqrt{73}}{4}
Discriminant D = 73 · x ≈ 3.386, -0.886001

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

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Graph of y = 2x² - 5x - 6-3-2-112345-10102030x = 3.3860009x = -0.88600094vertexy = 2x² - 5x - 6
Graph of y = 2x² - 5x - 6. The curve crosses the x-axis at x = 3.3860009 and x = -0.88600094; the vertex is at (1.25, -9.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -5, c = -6.

    2x2−5x−6=02x^2 - 5x - 6 = 0
  2. 2

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

    D=(−5)2−4⋅2⋅(−6)=25+48=73D = (-5)^2 - 4\cdot2\cdot(-6) = 25 + 48 = 73
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=5±734x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{73}}{4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=5+734orx=5−734x = \frac{5 + \sqrt{73}}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{73}}{4}
  6. 6

    As decimals: x ≈ 3.3860009 and x ≈ -0.88600094.

  7. 7

    Check numerically: for x ≈ 3.3860009 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 5x - 6 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{73}}{4},\ \frac{5 - \sqrt{73}}{4} \]
\[ 2x^2 - 5x - 6 = 0 \]
\[ D = (-5)^2 - 4\cdot2\cdot(-6) = 25 + 48 = 73 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{73}}{4} \]
\[ x = \frac{5 + \sqrt{73}}{4}\quad\text{or}\quad x = \frac{5 - \sqrt{73}}{4} \]

The parabola y = 2x² - 5x - 6

xy
-1.758.875
-0.75-1.125
0.25-7.125
1.25-9.125
2.25-7.125
3.25-1.125
4.258.875

Questions

What are the solutions of 2x² - 5x - 6 = 0?

The solutions of 2x² - 5x - 6 = 0 are x = (5 + √73)/4 or x = (5 - √73)/4, that is x ≈ 3.3860009 and x ≈ -0.88600094.

Does 2x² - 5x - 6 = 0 factor?

Not over the integers, because the discriminant 73 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 5x - 6?

(5/4, -73/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = -3.

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    Solve 2x² - 5x - 6 = 0: x = (5 + √73)/4 or x = (5 - √73)/4 (steps) | WordGPT