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Solve x² + 10x - 9 = 0

The solutions of x² + 10x - 9 = 0 are x = -5 + √34 or x = -5 - √34, that is x ≈ 0.83095189 and x ≈ -10.830952.

x² +x += 0
x2+10x−9=0⇒x=−5+34, −5−34x^2 + 10x - 9 = 0 \Rightarrow x = -5 + \sqrt{34},\ -5 - \sqrt{34}
Discriminant D = 136 · x ≈ 0.830952, -10.831

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

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Graph of y = x² + 10x - 9-12-10-8-6-4-22-40-30-20-1010203040x = 0.83095189x = -10.830952vertexy = x² + 10x - 9
Graph of y = x² + 10x - 9. The curve crosses the x-axis at x = 0.83095189 and x = -10.830952; the vertex is at (-5, -34).

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 = -9.

    x2+10x−9=0x^2 + 10x - 9 = 0
  2. 2

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

    D=102−4⋅1⋅(−9)=100+36=136D = 10^2 - 4\cdot1\cdot(-9) = 100 + 36 = 136
  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±1362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{136}}{2}
  5. 5

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

    x=−5+34orx=−5−34x = -5 + \sqrt{34}\quad\text{or}\quad x = -5 - \sqrt{34}
  6. 6

    As decimals: x ≈ 0.83095189 and x ≈ -10.830952.

  7. 7

    Check numerically: for x ≈ 0.83095189 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 10x - 9 = 0\ \Rightarrow\ x = -5 + \sqrt{34},\ -5 - \sqrt{34} \]
\[ x^2 + 10x - 9 = 0 \]
\[ D = 10^2 - 4\cdot1\cdot(-9) = 100 + 36 = 136 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{136}}{2} \]
\[ x = -5 + \sqrt{34}\quad\text{or}\quad x = -5 - \sqrt{34} \]

The parabola y = x² + 10x - 9

xy
-8-25
-7-30
-6-33
-5-34
-4-33
-3-30
-2-25

Questions

What are the solutions of x² + 10x - 9 = 0?

The solutions of x² + 10x - 9 = 0 are x = -5 + √34 or x = -5 - √34, that is x ≈ 0.83095189 and x ≈ -10.830952.

Does x² + 10x - 9 = 0 factor?

Not over the integers, because the discriminant 136 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 10x - 9?

(-5, -34). 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 = -9.

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    Solve x² + 10x - 9 = 0: x = -5 + √34 or x = -5 - √34 (steps) | WordGPT