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Solve 2x² + 9x - 3 = 0

The solutions of 2x² + 9x - 3 = 0 are x = (-9 + √105)/4 or x = (-9 - √105)/4, that is x ≈ 0.31173769 and x ≈ -4.8117377.

x² +x += 0
2x2+9x−3=0⇒x=−9+1054, −9−10542x^2 + 9x - 3 = 0 \Rightarrow x = \frac{-9 + \sqrt{105}}{4},\ \frac{-9 - \sqrt{105}}{4}
Discriminant D = 105 · x ≈ 0.311738, -4.81174

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

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Graph of y = 2x² + 9x - 3-7-6-5-4-3-2-112-1010203040x = 0.31173769x = -4.8117377vertexy = 2x² + 9x - 3
Graph of y = 2x² + 9x - 3. The curve crosses the x-axis at x = 0.31173769 and x = -4.8117377; the vertex is at (-2.25, -13.125).

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 = -3.

    2x2+9x−3=02x^2 + 9x - 3 = 0
  2. 2

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

    D=92−4⋅2⋅(−3)=81+24=105D = 9^2 - 4\cdot2\cdot(-3) = 81 + 24 = 105
  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±1054x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{105}}{4}
  5. 5

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

    x=−9+1054orx=−9−1054x = \frac{-9 + \sqrt{105}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{105}}{4}
  6. 6

    As decimals: x ≈ 0.31173769 and x ≈ -4.8117377.

  7. 7

    Check numerically: for x ≈ 0.31173769 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x - 3 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{105}}{4},\ \frac{-9 - \sqrt{105}}{4} \]
\[ 2x^2 + 9x - 3 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot(-3) = 81 + 24 = 105 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{105}}{4} \]
\[ x = \frac{-9 + \sqrt{105}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{105}}{4} \]

The parabola y = 2x² + 9x - 3

xy
-5.254.875
-4.25-5.125
-3.25-11.125
-2.25-13.125
-1.25-11.125
-0.25-5.125
0.754.875

Questions

What are the solutions of 2x² + 9x - 3 = 0?

The solutions of 2x² + 9x - 3 = 0 are x = (-9 + √105)/4 or x = (-9 - √105)/4, that is x ≈ 0.31173769 and x ≈ -4.8117377.

Does 2x² + 9x - 3 = 0 factor?

Not over the integers, because the discriminant 105 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 9x - 3?

(-9/4, -105/8). 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 = -3/2.

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    Solve 2x² + 9x - 3 = 0: x = (-9 + √105)/4 or x = (-9 - √105)/4 (steps) | WordGPT