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Solve 2x² - 6x - 3 = 0

The solutions of 2x² - 6x - 3 = 0 are x = (3 + √15)/2 or x = (3 - √15)/2, that is x ≈ 3.4364917 and x ≈ -0.43649167.

x² +x += 0
2x2−6x−3=0⇒x=3+152, 3−1522x^2 - 6x - 3 = 0 \Rightarrow x = \frac{3 + \sqrt{15}}{2},\ \frac{3 - \sqrt{15}}{2}
Discriminant D = 60 · x ≈ 3.43649, -0.436492

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

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Graph of y = 2x² - 6x - 3-2-112345-10102030x = 3.4364917x = -0.43649167vertexy = 2x² - 6x - 3
Graph of y = 2x² - 6x - 3. The curve crosses the x-axis at x = 3.4364917 and x = -0.43649167; the vertex is at (1.5, -7.5).

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 = -6, c = -3.

    2x2−6x−3=02x^2 - 6x - 3 = 0
  2. 2

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

    D=(−6)2−4⋅2⋅(−3)=36+24=60D = (-6)^2 - 4\cdot2\cdot(-3) = 36 + 24 = 60
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±604x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{60}}{4}
  5. 5

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

    x=3+152orx=3−152x = \frac{3 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{15}}{2}
  6. 6

    As decimals: x ≈ 3.4364917 and x ≈ -0.43649167.

  7. 7

    Check numerically: for x ≈ 3.4364917 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 6x - 3 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{15}}{2},\ \frac{3 - \sqrt{15}}{2} \]
\[ 2x^2 - 6x - 3 = 0 \]
\[ D = (-6)^2 - 4\cdot2\cdot(-3) = 36 + 24 = 60 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{60}}{4} \]
\[ x = \frac{3 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{15}}{2} \]

The parabola y = 2x² - 6x - 3

xy
-1.510.5
-0.50.5
0.5-5.5
1.5-7.5
2.5-5.5
3.50.5
4.510.5

Questions

What are the solutions of 2x² - 6x - 3 = 0?

The solutions of 2x² - 6x - 3 = 0 are x = (3 + √15)/2 or x = (3 - √15)/2, that is x ≈ 3.4364917 and x ≈ -0.43649167.

Does 2x² - 6x - 3 = 0 factor?

Not over the integers, because the discriminant 60 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 6x - 3?

(3/2, -15/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -3/2.

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    Solve 2x² - 6x - 3 = 0: x = (3 + √15)/2 or x = (3 - √15)/2 (steps) | WordGPT