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Solve 4x² - 6 = 0

The solutions of 4x² - 6 = 0 are x = (√6)/2 or x = (-√6)/2, that is x ≈ 1.2247449 and x ≈ -1.2247449.

x² +x += 0
4x2−6=0⇒x=62, −624x^2 - 6 = 0 \Rightarrow x = \frac{\sqrt{6}}{2},\ \frac{-\sqrt{6}}{2}
Discriminant D = 96 · x ≈ 1.22474, -1.22474

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

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Graph of y = 4x² - 6-3-2-1123-101020304050x = 1.2247449x = -1.2247449vertexy = 4x² - 6
Graph of y = 4x² - 6. The curve crosses the x-axis at x = 1.2247449 and x = -1.2247449; the vertex is at (0, -6).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = 0, c = -6.

    4x2−6=04x^2 - 6 = 0
  2. 2

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

    D=02−4⋅4⋅(−6)=0+96=96D = 0^2 - 4\cdot4\cdot(-6) = 0 + 96 = 96
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=0±968x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{96}}{8}
  5. 5

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

    x=62orx=−62x = \frac{\sqrt{6}}{2}\quad\text{or}\quad x = \frac{-\sqrt{6}}{2}
  6. 6

    As decimals: x ≈ 1.2247449 and x ≈ -1.2247449.

  7. 7

    Check numerically: for x ≈ 1.2247449 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 6 = 0\ \Rightarrow\ x = \frac{\sqrt{6}}{2},\ \frac{-\sqrt{6}}{2} \]
\[ 4x^2 - 6 = 0 \]
\[ D = 0^2 - 4\cdot4\cdot(-6) = 0 + 96 = 96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{96}}{8} \]
\[ x = \frac{\sqrt{6}}{2}\quad\text{or}\quad x = \frac{-\sqrt{6}}{2} \]

The parabola y = 4x² - 6

xy
-330
-210
-1-2
0-6
1-2
210
330

Questions

What are the solutions of 4x² - 6 = 0?

The solutions of 4x² - 6 = 0 are x = (√6)/2 or x = (-√6)/2, that is x ≈ 1.2247449 and x ≈ -1.2247449.

Does 4x² - 6 = 0 factor?

Not over the integers, because the discriminant 96 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 4x² - 6?

(0, -6). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -3/2.

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