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Solve 2x² + 9x - 2 = 0

The solutions of 2x² + 9x - 2 = 0 are x = (-9 + √97)/4 or x = (-9 - √97)/4, that is x ≈ 0.21221445 and x ≈ -4.7122145.

x² +x += 0
2x2+9x−2=0⇒x=−9+974, −9−9742x^2 + 9x - 2 = 0 \Rightarrow x = \frac{-9 + \sqrt{97}}{4},\ \frac{-9 - \sqrt{97}}{4}
Discriminant D = 97 · x ≈ 0.212214, -4.71221

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

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Graph of y = 2x² + 9x - 2-7-6-5-4-3-2-112-1010203040x = 0.21221445x = -4.7122145vertexy = 2x² + 9x - 2
Graph of y = 2x² + 9x - 2. The curve crosses the x-axis at x = 0.21221445 and x = -4.7122145; the vertex is at (-2.25, -12.125).

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 = 9, c = -2.

    2x2+9x−2=02x^2 + 9x - 2 = 0
  2. 2

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

    D=92−4⋅2⋅(−2)=81+16=97D = 9^2 - 4\cdot2\cdot(-2) = 81 + 16 = 97
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±974x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{97}}{4}
  5. 5

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

    x=−9+974orx=−9−974x = \frac{-9 + \sqrt{97}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{97}}{4}
  6. 6

    As decimals: x ≈ 0.21221445 and x ≈ -4.7122145.

  7. 7

    Check numerically: for x ≈ 0.21221445 the left side is -1.9984014e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x - 2 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{97}}{4},\ \frac{-9 - \sqrt{97}}{4} \]
\[ 2x^2 + 9x - 2 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot(-2) = 81 + 16 = 97 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{97}}{4} \]
\[ x = \frac{-9 + \sqrt{97}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{97}}{4} \]

The parabola y = 2x² + 9x - 2

xy
-5.255.875
-4.25-4.125
-3.25-10.125
-2.25-12.125
-1.25-10.125
-0.25-4.125
0.755.875

Questions

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

The solutions of 2x² + 9x - 2 = 0 are x = (-9 + √97)/4 or x = (-9 - √97)/4, that is x ≈ 0.21221445 and x ≈ -4.7122145.

Does 2x² + 9x - 2 = 0 factor?

Not over the integers, because the discriminant 97 is not a perfect square. Use the quadratic formula.

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

(-9/4, -97/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = -1.

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    Solve 2x² + 9x - 2 = 0: x = (-9 + √97)/4 or x = (-9 - √97)/4 (steps) | WordGPT