WordGPT

Solve 2x² + 5x - 5 = 0

The solutions of 2x² + 5x - 5 = 0 are x = (-5 + √65)/4 or x = (-5 - √65)/4, that is x ≈ 0.76556444 and x ≈ -3.2655644.

x² +x += 0
2x2+5x−5=0⇒x=−5+654, −5−6542x^2 + 5x - 5 = 0 \Rightarrow x = \frac{-5 + \sqrt{65}}{4},\ \frac{-5 - \sqrt{65}}{4}
Discriminant D = 65 · x ≈ 0.765564, -3.26556

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

Explain it step by step in chatMake a short video of it
Graph of y = 2x² + 5x - 5-5-4-3-2-1123-10102030x = 0.76556444x = -3.2655644vertexy = 2x² + 5x - 5
Graph of y = 2x² + 5x - 5. The curve crosses the x-axis at x = 0.76556444 and x = -3.2655644; the vertex is at (-1.25, -8.125).

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 = 5, c = -5.

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

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

    D=52−4⋅2⋅(−5)=25+40=65D = 5^2 - 4\cdot2\cdot(-5) = 25 + 40 = 65
  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±654x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{65}}{4}
  5. 5

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

    x=−5+654orx=−5−654x = \frac{-5 + \sqrt{65}}{4}\quad\text{or}\quad x = \frac{-5 - \sqrt{65}}{4}
  6. 6

    As decimals: x ≈ 0.76556444 and x ≈ -3.2655644.

  7. 7

    Check numerically: for x ≈ 0.76556444 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 5x - 5 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{65}}{4},\ \frac{-5 - \sqrt{65}}{4} \]
\[ 2x^2 + 5x - 5 = 0 \]
\[ D = 5^2 - 4\cdot2\cdot(-5) = 25 + 40 = 65 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{65}}{4} \]
\[ x = \frac{-5 + \sqrt{65}}{4}\quad\text{or}\quad x = \frac{-5 - \sqrt{65}}{4} \]

The parabola y = 2x² + 5x - 5

xy
-4.259.875
-3.25-0.125
-2.25-6.125
-1.25-8.125
-0.25-6.125
0.75-0.125
1.759.875

Questions

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

The solutions of 2x² + 5x - 5 = 0 are x = (-5 + √65)/4 or x = (-5 - √65)/4, that is x ≈ 0.76556444 and x ≈ -3.2655644.

Does 2x² + 5x - 5 = 0 factor?

Not over the integers, because the discriminant 65 is not a perfect square. Use the quadratic formula.

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

(-5/4, -65/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

Similar equations

    Solve 2x² + 5x - 5 = 0: x = (-5 + √65)/4 or x = (-5 - √65)/4 (steps) | WordGPT