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Solve x² - 7x - 3 = 0

The solutions of x² - 7x - 3 = 0 are x = (7 + √61)/2 or x = (7 - √61)/2, that is x ≈ 7.4051248 and x ≈ -0.40512484.

x² +x += 0
x2−7x−3=0⇒x=7+612, 7−612x^2 - 7x - 3 = 0 \Rightarrow x = \frac{7 + \sqrt{61}}{2},\ \frac{7 - \sqrt{61}}{2}
Discriminant D = 61 · x ≈ 7.40512, -0.405125

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

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Graph of y = x² - 7x - 3-22468-20-101020x = 7.4051248x = -0.40512484vertexy = x² - 7x - 3
Graph of y = x² - 7x - 3. The curve crosses the x-axis at x = 7.4051248 and x = -0.40512484; the vertex is at (3.5, -15.25).

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 = -7, c = -3.

    x2−7x−3=0x^2 - 7x - 3 = 0
  2. 2

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

    D=(−7)2−4⋅1⋅(−3)=49+12=61D = (-7)^2 - 4\cdot1\cdot(-3) = 49 + 12 = 61
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=7±612x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{61}}{2}
  5. 5

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

    x=7+612orx=7−612x = \frac{7 + \sqrt{61}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{61}}{2}
  6. 6

    As decimals: x ≈ 7.4051248 and x ≈ -0.40512484.

  7. 7

    Check numerically: for x ≈ 7.4051248 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 7x - 3 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{61}}{2},\ \frac{7 - \sqrt{61}}{2} \]
\[ x^2 - 7x - 3 = 0 \]
\[ D = (-7)^2 - 4\cdot1\cdot(-3) = 49 + 12 = 61 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{61}}{2} \]
\[ x = \frac{7 + \sqrt{61}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{61}}{2} \]

The parabola y = x² - 7x - 3

xy
0.5-6.25
1.5-11.25
2.5-14.25
3.5-15.25
4.5-14.25
5.5-11.25
6.5-6.25

Questions

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

The solutions of x² - 7x - 3 = 0 are x = (7 + √61)/2 or x = (7 - √61)/2, that is x ≈ 7.4051248 and x ≈ -0.40512484.

Does x² - 7x - 3 = 0 factor?

Not over the integers, because the discriminant 61 is not a perfect square. Use the quadratic formula.

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

(7/2, -61/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7, product = c/a = -3.

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    Solve x² - 7x - 3 = 0: x = (7 + √61)/2 or x = (7 - √61)/2 (steps) | WordGPT