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Solve x² + 7x - 12 = 0

The solutions of x² + 7x - 12 = 0 are x = (-7 + √97)/2 or x = (-7 - √97)/2, that is x ≈ 1.4244289 and x ≈ -8.4244289.

x² +x += 0
x2+7x−12=0⇒x=−7+972, −7−972x^2 + 7x - 12 = 0 \Rightarrow x = \frac{-7 + \sqrt{97}}{2},\ \frac{-7 - \sqrt{97}}{2}
Discriminant D = 97 · x ≈ 1.42443, -8.42443

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

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Graph of y = x² + 7x - 12-10-8-6-4-22-30-20-10102030x = 1.4244289x = -8.4244289vertexy = x² + 7x - 12
Graph of y = x² + 7x - 12. The curve crosses the x-axis at x = 1.4244289 and x = -8.4244289; the vertex is at (-3.5, -24.25).

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 = 7, c = -12.

    x2+7x−12=0x^2 + 7x - 12 = 0
  2. 2

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

    D=72−4⋅1⋅(−12)=49+48=97D = 7^2 - 4\cdot1\cdot(-12) = 49 + 48 = 97
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−7±972x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{97}}{2}
  5. 5

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

    x=−7+972orx=−7−972x = \frac{-7 + \sqrt{97}}{2}\quad\text{or}\quad x = \frac{-7 - \sqrt{97}}{2}
  6. 6

    As decimals: x ≈ 1.4244289 and x ≈ -8.4244289.

  7. 7

    Check numerically: for x ≈ 1.4244289 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 7x - 12 = 0\ \Rightarrow\ x = \frac{-7 + \sqrt{97}}{2},\ \frac{-7 - \sqrt{97}}{2} \]
\[ x^2 + 7x - 12 = 0 \]
\[ D = 7^2 - 4\cdot1\cdot(-12) = 49 + 48 = 97 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{97}}{2} \]
\[ x = \frac{-7 + \sqrt{97}}{2}\quad\text{or}\quad x = \frac{-7 - \sqrt{97}}{2} \]

The parabola y = x² + 7x - 12

xy
-6.5-15.25
-5.5-20.25
-4.5-23.25
-3.5-24.25
-2.5-23.25
-1.5-20.25
-0.5-15.25

Questions

What are the solutions of x² + 7x - 12 = 0?

The solutions of x² + 7x - 12 = 0 are x = (-7 + √97)/2 or x = (-7 - √97)/2, that is x ≈ 1.4244289 and x ≈ -8.4244289.

Does x² + 7x - 12 = 0 factor?

Not over the integers, because the discriminant 97 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 7x - 12?

(-7/2, -97/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7, product = c/a = -12.

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    Solve x² + 7x - 12 = 0: x = (-7 + √97)/2 or x = (-7 - √97)/2 (steps) | WordGPT