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Solve 2x² + 6x - 7 = 0

The solutions of 2x² + 6x - 7 = 0 are x = (-3 + √23)/2 or x = (-3 - √23)/2, that is x ≈ 0.89791576 and x ≈ -3.8979158.

x² +x += 0
2x2+6x−7=0⇒x=−3+232, −3−2322x^2 + 6x - 7 = 0 \Rightarrow x = \frac{-3 + \sqrt{23}}{2},\ \frac{-3 - \sqrt{23}}{2}
Discriminant D = 92 · x ≈ 0.897916, -3.89792

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

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Graph of y = 2x² + 6x - 7-6-5-4-3-2-1123-1010203040x = 0.89791576x = -3.8979158vertexy = 2x² + 6x - 7
Graph of y = 2x² + 6x - 7. The curve crosses the x-axis at x = 0.89791576 and x = -3.8979158; the vertex is at (-1.5, -11.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = 6, c = -7.

    2x2+6x−7=02x^2 + 6x - 7 = 0
  2. 2

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

    D=62−4⋅2⋅(−7)=36+56=92D = 6^2 - 4\cdot2\cdot(-7) = 36 + 56 = 92
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±924x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{92}}{4}
  5. 5

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

    x=−3+232orx=−3−232x = \frac{-3 + \sqrt{23}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{23}}{2}
  6. 6

    As decimals: x ≈ 0.89791576 and x ≈ -3.8979158.

  7. 7

    Check numerically: for x ≈ 0.89791576 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 6x - 7 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{23}}{2},\ \frac{-3 - \sqrt{23}}{2} \]
\[ 2x^2 + 6x - 7 = 0 \]
\[ D = 6^2 - 4\cdot2\cdot(-7) = 36 + 56 = 92 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{92}}{4} \]
\[ x = \frac{-3 + \sqrt{23}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{23}}{2} \]

The parabola y = 2x² + 6x - 7

xy
-4.56.5
-3.5-3.5
-2.5-9.5
-1.5-11.5
-0.5-9.5
0.5-3.5
1.56.5

Questions

What are the solutions of 2x² + 6x - 7 = 0?

The solutions of 2x² + 6x - 7 = 0 are x = (-3 + √23)/2 or x = (-3 - √23)/2, that is x ≈ 0.89791576 and x ≈ -3.8979158.

Does 2x² + 6x - 7 = 0 factor?

Not over the integers, because the discriminant 92 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 6x - 7?

(-3/2, -23/2). 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 = -7/2.

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    Solve 2x² + 6x - 7 = 0: x = (-3 + √23)/2 or x = (-3 - √23)/2 (steps) | WordGPT