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Solve x² - 3x - 7 = 0

The solutions of x² - 3x - 7 = 0 are x = (3 + √37)/2 or x = (3 - √37)/2, that is x ≈ 4.5413813 and x ≈ -1.5413813.

x² +x += 0
x2−3x−7=0⇒x=3+372, 3−372x^2 - 3x - 7 = 0 \Rightarrow x = \frac{3 + \sqrt{37}}{2},\ \frac{3 - \sqrt{37}}{2}
Discriminant D = 37 · x ≈ 4.54138, -1.54138

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

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Graph of y = x² - 3x - 7-4-2246-10-55101520x = 4.5413813x = -1.5413813vertexy = x² - 3x - 7
Graph of y = x² - 3x - 7. The curve crosses the x-axis at x = 4.5413813 and x = -1.5413813; the vertex is at (1.5, -9.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -3, c = -7.

    x2−3x−7=0x^2 - 3x - 7 = 0
  2. 2

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

    D=(−3)2−4⋅1⋅(−7)=9+28=37D = (-3)^2 - 4\cdot1\cdot(-7) = 9 + 28 = 37
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±372x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{37}}{2}
  5. 5

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

    x=3+372orx=3−372x = \frac{3 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{37}}{2}
  6. 6

    As decimals: x ≈ 4.5413813 and x ≈ -1.5413813.

  7. 7

    Check numerically: for x ≈ 4.5413813 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 3x - 7 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{37}}{2},\ \frac{3 - \sqrt{37}}{2} \]
\[ x^2 - 3x - 7 = 0 \]
\[ D = (-3)^2 - 4\cdot1\cdot(-7) = 9 + 28 = 37 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{37}}{2} \]
\[ x = \frac{3 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{37}}{2} \]

The parabola y = x² - 3x - 7

xy
-1.5-0.25
-0.5-5.25
0.5-8.25
1.5-9.25
2.5-8.25
3.5-5.25
4.5-0.25

Questions

What are the solutions of x² - 3x - 7 = 0?

The solutions of x² - 3x - 7 = 0 are x = (3 + √37)/2 or x = (3 - √37)/2, that is x ≈ 4.5413813 and x ≈ -1.5413813.

Does x² - 3x - 7 = 0 factor?

Not over the integers, because the discriminant 37 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 3x - 7?

(3/2, -37/4). 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 = -7.

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    Solve x² - 3x - 7 = 0: x = (3 + √37)/2 or x = (3 - √37)/2 (steps) | WordGPT