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Solve x² + 5x - 12 = 0

The solutions of x² + 5x - 12 = 0 are x = (-5 + √73)/2 or x = (-5 - √73)/2, that is x ≈ 1.7720019 and x ≈ -6.7720019.

x² +x += 0
x2+5x−12=0⇒x=−5+732, −5−732x^2 + 5x - 12 = 0 \Rightarrow x = \frac{-5 + \sqrt{73}}{2},\ \frac{-5 - \sqrt{73}}{2}
Discriminant D = 73 · x ≈ 1.772, -6.772

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

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Graph of y = x² + 5x - 12-8-6-4-224-20-10102030x = 1.7720019x = -6.7720019vertexy = x² + 5x - 12
Graph of y = x² + 5x - 12. The curve crosses the x-axis at x = 1.7720019 and x = -6.7720019; the vertex is at (-2.5, -18.25).

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 = 5, c = -12.

    x2+5x−12=0x^2 + 5x - 12 = 0
  2. 2

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

    D=52−4⋅1⋅(−12)=25+48=73D = 5^2 - 4\cdot1\cdot(-12) = 25 + 48 = 73
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−5±732x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{73}}{2}
  5. 5

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

    x=−5+732orx=−5−732x = \frac{-5 + \sqrt{73}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{73}}{2}
  6. 6

    As decimals: x ≈ 1.7720019 and x ≈ -6.7720019.

  7. 7

    Check numerically: for x ≈ 1.7720019 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 5x - 12 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{73}}{2},\ \frac{-5 - \sqrt{73}}{2} \]
\[ x^2 + 5x - 12 = 0 \]
\[ D = 5^2 - 4\cdot1\cdot(-12) = 25 + 48 = 73 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{73}}{2} \]
\[ x = \frac{-5 + \sqrt{73}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{73}}{2} \]

The parabola y = x² + 5x - 12

xy
-5.5-9.25
-4.5-14.25
-3.5-17.25
-2.5-18.25
-1.5-17.25
-0.5-14.25
0.5-9.25

Questions

What are the solutions of x² + 5x - 12 = 0?

The solutions of x² + 5x - 12 = 0 are x = (-5 + √73)/2 or x = (-5 - √73)/2, that is x ≈ 1.7720019 and x ≈ -6.7720019.

Does x² + 5x - 12 = 0 factor?

Not over the integers, because the discriminant 73 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 5x - 12?

(-5/2, -73/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = -12.

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    Solve x² + 5x - 12 = 0: x = (-5 + √73)/2 or x = (-5 - √73)/2 (steps) | WordGPT