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Solve 2x² + 7x - 5 = 0

The solutions of 2x² + 7x - 5 = 0 are x = (-7 + √89)/4 or x = (-7 - √89)/4, that is x ≈ 0.60849528 and x ≈ -4.1084953.

x² +x += 0
2x2+7x−5=0⇒x=−7+894, −7−8942x^2 + 7x - 5 = 0 \Rightarrow x = \frac{-7 + \sqrt{89}}{4},\ \frac{-7 - \sqrt{89}}{4}
Discriminant D = 89 · x ≈ 0.608495, -4.1085

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

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Graph of y = 2x² + 7x - 5-6-5-4-3-2-1123-1010203040x = 0.60849528x = -4.1084953vertexy = 2x² + 7x - 5
Graph of y = 2x² + 7x - 5. The curve crosses the x-axis at x = 0.60849528 and x = -4.1084953; the vertex is at (-1.75, -11.125).

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 = 7, c = -5.

    2x2+7x−5=02x^2 + 7x - 5 = 0
  2. 2

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

    D=72−4⋅2⋅(−5)=49+40=89D = 7^2 - 4\cdot2\cdot(-5) = 49 + 40 = 89
  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±894x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{89}}{4}
  5. 5

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

    x=−7+894orx=−7−894x = \frac{-7 + \sqrt{89}}{4}\quad\text{or}\quad x = \frac{-7 - \sqrt{89}}{4}
  6. 6

    As decimals: x ≈ 0.60849528 and x ≈ -4.1084953.

  7. 7

    Check numerically: for x ≈ 0.60849528 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 7x - 5 = 0\ \Rightarrow\ x = \frac{-7 + \sqrt{89}}{4},\ \frac{-7 - \sqrt{89}}{4} \]
\[ 2x^2 + 7x - 5 = 0 \]
\[ D = 7^2 - 4\cdot2\cdot(-5) = 49 + 40 = 89 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{89}}{4} \]
\[ x = \frac{-7 + \sqrt{89}}{4}\quad\text{or}\quad x = \frac{-7 - \sqrt{89}}{4} \]

The parabola y = 2x² + 7x - 5

xy
-4.756.875
-3.75-3.125
-2.75-9.125
-1.75-11.125
-0.75-9.125
0.25-3.125
1.256.875

Questions

What are the solutions of 2x² + 7x - 5 = 0?

The solutions of 2x² + 7x - 5 = 0 are x = (-7 + √89)/4 or x = (-7 - √89)/4, that is x ≈ 0.60849528 and x ≈ -4.1084953.

Does 2x² + 7x - 5 = 0 factor?

Not over the integers, because the discriminant 89 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 7x - 5?

(-7/4, -89/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/2, product = c/a = -5/2.

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    Solve 2x² + 7x - 5 = 0: x = (-7 + √89)/4 or x = (-7 - √89)/4 (steps) | WordGPT