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Solve 4x² - 12x - 8 = 0

The solutions of 4x² - 12x - 8 = 0 are x = (3 + √17)/2 or x = (3 - √17)/2, that is x ≈ 3.5615528 and x ≈ -0.56155281.

x² +x += 0
4x2−12x−8=0⇒x=3+172, 3−1724x^2 - 12x - 8 = 0 \Rightarrow x = \frac{3 + \sqrt{17}}{2},\ \frac{3 - \sqrt{17}}{2}
Discriminant D = 272 · x ≈ 3.56155, -0.561553

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

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Graph of y = 4x² - 12x - 8-3-2-1123456-20204060x = 3.5615528x = -0.56155281vertexy = 4x² - 12x - 8
Graph of y = 4x² - 12x - 8. The curve crosses the x-axis at x = 3.5615528 and x = -0.56155281; the vertex is at (1.5, -17).

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 = -12, c = -8.

    4x2−12x−8=04x^2 - 12x - 8 = 0
  2. 2

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

    D=(−12)2−4⋅4⋅(−8)=144+128=272D = (-12)^2 - 4\cdot4\cdot(-8) = 144 + 128 = 272
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±2728x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{272}}{8}
  5. 5

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

    x=3+172orx=3−172x = \frac{3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{17}}{2}
  6. 6

    As decimals: x ≈ 3.5615528 and x ≈ -0.56155281.

  7. 7

    Check numerically: for x ≈ 3.5615528 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 12x - 8 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{17}}{2},\ \frac{3 - \sqrt{17}}{2} \]
\[ 4x^2 - 12x - 8 = 0 \]
\[ D = (-12)^2 - 4\cdot4\cdot(-8) = 144 + 128 = 272 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{272}}{8} \]
\[ x = \frac{3 + \sqrt{17}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{17}}{2} \]

The parabola y = 4x² - 12x - 8

xy
-1.519
-0.5-1
0.5-13
1.5-17
2.5-13
3.5-1
4.519

Questions

What are the solutions of 4x² - 12x - 8 = 0?

The solutions of 4x² - 12x - 8 = 0 are x = (3 + √17)/2 or x = (3 - √17)/2, that is x ≈ 3.5615528 and x ≈ -0.56155281.

Does 4x² - 12x - 8 = 0 factor?

Not over the integers, because the discriminant 272 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 4x² - 12x - 8?

(3/2, -17). 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 = -2.

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    Solve 4x² - 12x - 8 = 0: x = (3 + √17)/2 or x = (3 - √17)/2 (steps) | WordGPT