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Solve -x² + 10x + 12 = 0

The solutions of -x² + 10x + 12 = 0 are x = 5 + √37 or x = 5 - √37, that is x ≈ -1.0827625 and x ≈ 11.082763.

x² +x += 0
−x2+10x+12=0⇒x=5+37, 5−37-x^2 + 10x + 12 = 0 \Rightarrow x = 5 + \sqrt{37},\ 5 - \sqrt{37}
Discriminant D = 148 · x ≈ -1.08276, 11.0828

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

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Graph of y = -x² + 10x + 12-224681012-40-30-20-1010203040x = -1.0827625x = 11.082763vertexy = -x² + 10x + 12
Graph of y = -x² + 10x + 12. The curve crosses the x-axis at x = -1.0827625 and x = 11.082763; the vertex is at (5, 37).

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 = 12.

    −x2+10x+12=0-x^2 + 10x + 12 = 0
  2. 2

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

    D=102−4⋅(−1)⋅12=100+48=148D = 10^2 - 4\cdot(-1)\cdot12 = 100 + 48 = 148
  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±148−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{148}}{-2}
  5. 5

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

    x=5+37orx=5−37x = 5 + \sqrt{37}\quad\text{or}\quad x = 5 - \sqrt{37}
  6. 6

    As decimals: x ≈ -1.0827625 and x ≈ 11.082763.

  7. 7

    Check numerically: for x ≈ -1.0827625 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x + 12 = 0\ \Rightarrow\ x = 5 + \sqrt{37},\ 5 - \sqrt{37} \]
\[ -x^2 + 10x + 12 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot12 = 100 + 48 = 148 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{148}}{-2} \]
\[ x = 5 + \sqrt{37}\quad\text{or}\quad x = 5 - \sqrt{37} \]

The parabola y = -x² + 10x + 12

xy
228
333
436
537
636
733
828

Questions

What are the solutions of -x² + 10x + 12 = 0?

The solutions of -x² + 10x + 12 = 0 are x = 5 + √37 or x = 5 - √37, that is x ≈ -1.0827625 and x ≈ 11.082763.

Does -x² + 10x + 12 = 0 factor?

Not over the integers, because the discriminant 148 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 10x + 12?

(5, 37). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = -12.

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    Solve -x² + 10x + 12 = 0: x = 5 + √37 or x = 5 - √37 (steps) | WordGPT