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Solve x² + 3x - 7 = 0

The solutions of x² + 3x - 7 = 0 are x = (-3 + √37)/2 or x = (-3 - √37)/2, that is x ≈ 1.5413813 and x ≈ -4.5413813.

x² +x += 0
x2+3x−7=0⇒x=−3+372, −3−372x^2 + 3x - 7 = 0 \Rightarrow x = \frac{-3 + \sqrt{37}}{2},\ \frac{-3 - \sqrt{37}}{2}
Discriminant D = 37 · x ≈ 1.54138, -4.54138

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

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Graph of y = x² + 3x - 7-6-4-224-10-55101520x = 1.5413813x = -4.5413813vertexy = x² + 3x - 7
Graph of y = x² + 3x - 7. The curve crosses the x-axis at x = 1.5413813 and x = -4.5413813; the vertex is at (-1.5, -9.25).

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 = 3, c = -7.

    x2+3x−7=0x^2 + 3x - 7 = 0
  2. 2

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

    D=32−4⋅1⋅(−7)=9+28=37D = 3^2 - 4\cdot1\cdot(-7) = 9 + 28 = 37
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±372x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{37}}{2}
  5. 5

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

    x=−3+372orx=−3−372x = \frac{-3 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{37}}{2}
  6. 6

    As decimals: x ≈ 1.5413813 and x ≈ -4.5413813.

  7. 7

    Check numerically: for x ≈ 1.5413813 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 3x - 7 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{37}}{2},\ \frac{-3 - \sqrt{37}}{2} \]
\[ x^2 + 3x - 7 = 0 \]
\[ D = 3^2 - 4\cdot1\cdot(-7) = 9 + 28 = 37 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{37}}{2} \]
\[ x = \frac{-3 + \sqrt{37}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{37}}{2} \]

The parabola y = x² + 3x - 7

xy
-4.5-0.25
-3.5-5.25
-2.5-8.25
-1.5-9.25
-0.5-8.25
0.5-5.25
1.5-0.25

Questions

What are the solutions of x² + 3x - 7 = 0?

The solutions of x² + 3x - 7 = 0 are x = (-3 + √37)/2 or x = (-3 - √37)/2, that is x ≈ 1.5413813 and x ≈ -4.5413813.

Does x² + 3x - 7 = 0 factor?

Not over the integers, because the discriminant 37 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 3x - 7?

(-3/2, -37/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -7.

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    Solve x² + 3x - 7 = 0: x = (-3 + √37)/2 or x = (-3 - √37)/2 (steps) | WordGPT