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Solve x² - 10x - 3 = 0

The solutions of x² - 10x - 3 = 0 are x = 5 + 2√7 or x = 5 - 2√7, that is x ≈ 10.291503 and x ≈ -0.29150262.

x² +x += 0
x2−10x−3=0⇒x=5+27, 5−27x^2 - 10x - 3 = 0 \Rightarrow x = 5 + 2\sqrt{7},\ 5 - 2\sqrt{7}
Discriminant D = 112 · x ≈ 10.2915, -0.291503

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

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Graph of y = x² - 10x - 3-224681012-30-20-10102030x = 10.291503x = -0.29150262vertexy = x² - 10x - 3
Graph of y = x² - 10x - 3. The curve crosses the x-axis at x = 10.291503 and x = -0.29150262; the vertex is at (5, -28).

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 = -10, c = -3.

    x2−10x−3=0x^2 - 10x - 3 = 0
  2. 2

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

    D=(−10)2−4⋅1⋅(−3)=100+12=112D = (-10)^2 - 4\cdot1\cdot(-3) = 100 + 12 = 112
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=10±1122x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{112}}{2}
  5. 5

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

    x=5+27orx=5−27x = 5 + 2\sqrt{7}\quad\text{or}\quad x = 5 - 2\sqrt{7}
  6. 6

    As decimals: x ≈ 10.291503 and x ≈ -0.29150262.

  7. 7

    Check numerically: for x ≈ 10.291503 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 3 = 0\ \Rightarrow\ x = 5 + 2\sqrt{7},\ 5 - 2\sqrt{7} \]
\[ x^2 - 10x - 3 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-3) = 100 + 12 = 112 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{112}}{2} \]
\[ x = 5 + 2\sqrt{7}\quad\text{or}\quad x = 5 - 2\sqrt{7} \]

The parabola y = x² - 10x - 3

xy
2-19
3-24
4-27
5-28
6-27
7-24
8-19

Questions

What are the solutions of x² - 10x - 3 = 0?

The solutions of x² - 10x - 3 = 0 are x = 5 + 2√7 or x = 5 - 2√7, that is x ≈ 10.291503 and x ≈ -0.29150262.

Does x² - 10x - 3 = 0 factor?

Not over the integers, because the discriminant 112 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x - 3?

(5, -28). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = -3.

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