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Solve 4x² - 4x - 9 = 0

The solutions of 4x² - 4x - 9 = 0 are x = (1 + √10)/2 or x = (1 - √10)/2, that is x ≈ 2.0811388 and x ≈ -1.0811388.

x² +x += 0
4x2−4x−9=0⇒x=1+102, 1−1024x^2 - 4x - 9 = 0 \Rightarrow x = \frac{1 + \sqrt{10}}{2},\ \frac{1 - \sqrt{10}}{2}
Discriminant D = 160 · x ≈ 2.08114, -1.08114

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

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Graph of y = 4x² - 4x - 9-3-2-11234-10102030405060x = 2.0811388x = -1.0811388vertexy = 4x² - 4x - 9
Graph of y = 4x² - 4x - 9. The curve crosses the x-axis at x = 2.0811388 and x = -1.0811388; the vertex is at (0.5, -10).

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 = -4, c = -9.

    4x2−4x−9=04x^2 - 4x - 9 = 0
  2. 2

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

    D=(−4)2−4⋅4⋅(−9)=16+144=160D = (-4)^2 - 4\cdot4\cdot(-9) = 16 + 144 = 160
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=4±1608x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{160}}{8}
  5. 5

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

    x=1+102orx=1−102x = \frac{1 + \sqrt{10}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{10}}{2}
  6. 6

    As decimals: x ≈ 2.0811388 and x ≈ -1.0811388.

  7. 7

    Check numerically: for x ≈ 2.0811388 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 4x - 9 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{10}}{2},\ \frac{1 - \sqrt{10}}{2} \]
\[ 4x^2 - 4x - 9 = 0 \]
\[ D = (-4)^2 - 4\cdot4\cdot(-9) = 16 + 144 = 160 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{160}}{8} \]
\[ x = \frac{1 + \sqrt{10}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{10}}{2} \]

The parabola y = 4x² - 4x - 9

xy
-2.526
-1.56
-0.5-6
0.5-10
1.5-6
2.56
3.526

Questions

What are the solutions of 4x² - 4x - 9 = 0?

The solutions of 4x² - 4x - 9 = 0 are x = (1 + √10)/2 or x = (1 - √10)/2, that is x ≈ 2.0811388 and x ≈ -1.0811388.

Does 4x² - 4x - 9 = 0 factor?

Not over the integers, because the discriminant 160 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 4x² - 4x - 9?

(1/2, -10). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -9/4.

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