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Solve x² + 11x - 3 = 0

The solutions of x² + 11x - 3 = 0 are x = (-11 + √133)/2 or x = (-11 - √133)/2, that is x ≈ 0.2662813 and x ≈ -11.266281.

x² +x += 0
x2+11x−3=0⇒x=−11+1332, −11−1332x^2 + 11x - 3 = 0 \Rightarrow x = \frac{-11 + \sqrt{133}}{2},\ \frac{-11 - \sqrt{133}}{2}
Discriminant D = 133 · x ≈ 0.266281, -11.2663

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

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Graph of y = x² + 11x - 3-12-10-8-6-4-22-40-30-20-1010203040x = 0.2662813x = -11.266281vertexy = x² + 11x - 3
Graph of y = x² + 11x - 3. The curve crosses the x-axis at x = 0.2662813 and x = -11.266281; the vertex is at (-5.5, -33.25).

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 = 11, c = -3.

    x2+11x−3=0x^2 + 11x - 3 = 0
  2. 2

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

    D=112−4⋅1⋅(−3)=121+12=133D = 11^2 - 4\cdot1\cdot(-3) = 121 + 12 = 133
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1332x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{133}}{2}
  5. 5

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

    x=−11+1332orx=−11−1332x = \frac{-11 + \sqrt{133}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{133}}{2}
  6. 6

    As decimals: x ≈ 0.2662813 and x ≈ -11.266281.

  7. 7

    Check numerically: for x ≈ 0.2662813 the left side is 3.9968029e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 11x - 3 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{133}}{2},\ \frac{-11 - \sqrt{133}}{2} \]
\[ x^2 + 11x - 3 = 0 \]
\[ D = 11^2 - 4\cdot1\cdot(-3) = 121 + 12 = 133 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{133}}{2} \]
\[ x = \frac{-11 + \sqrt{133}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{133}}{2} \]

The parabola y = x² + 11x - 3

xy
-8.5-24.25
-7.5-29.25
-6.5-32.25
-5.5-33.25
-4.5-32.25
-3.5-29.25
-2.5-24.25

Questions

What are the solutions of x² + 11x - 3 = 0?

The solutions of x² + 11x - 3 = 0 are x = (-11 + √133)/2 or x = (-11 - √133)/2, that is x ≈ 0.2662813 and x ≈ -11.266281.

Does x² + 11x - 3 = 0 factor?

Not over the integers, because the discriminant 133 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 11x - 3?

(-11/2, -133/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11, product = c/a = -3.

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    Solve x² + 11x - 3 = 0: x = (-11 + √133)/2 or x = (-11 - √133)/2 (steps) | WordGPT