WordGPT

Solve 2x² + 9x - 8 = 0

The solutions of 2x² + 9x - 8 = 0 are x = (-9 + √145)/4 or x = (-9 - √145)/4, that is x ≈ 0.76039864 and x ≈ -5.2603986.

x² +x += 0
2x2+9x−8=0⇒x=−9+1454, −9−14542x^2 + 9x - 8 = 0 \Rightarrow x = \frac{-9 + \sqrt{145}}{4},\ \frac{-9 - \sqrt{145}}{4}
Discriminant D = 145 · x ≈ 0.760399, -5.2604

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 + 9x - 8-6-4-22-20-1010203040x = 0.76039864x = -5.2603986vertexy = 2x² + 9x - 8
Graph of y = 2x² + 9x - 8. The curve crosses the x-axis at x = 0.76039864 and x = -5.2603986; the vertex is at (-2.25, -18.125).

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 = 9, c = -8.

    2x2+9x−8=02x^2 + 9x - 8 = 0
  2. 2

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

    D=92−4⋅2⋅(−8)=81+64=145D = 9^2 - 4\cdot2\cdot(-8) = 81 + 64 = 145
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±1454x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{145}}{4}
  5. 5

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

    x=−9+1454orx=−9−1454x = \frac{-9 + \sqrt{145}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{145}}{4}
  6. 6

    As decimals: x ≈ 0.76039864 and x ≈ -5.2603986.

  7. 7

    Check numerically: for x ≈ 0.76039864 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x - 8 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{145}}{4},\ \frac{-9 - \sqrt{145}}{4} \]
\[ 2x^2 + 9x - 8 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot(-8) = 81 + 64 = 145 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{145}}{4} \]
\[ x = \frac{-9 + \sqrt{145}}{4}\quad\text{or}\quad x = \frac{-9 - \sqrt{145}}{4} \]

The parabola y = 2x² + 9x - 8

xy
-5.25-0.125
-4.25-10.125
-3.25-16.125
-2.25-18.125
-1.25-16.125
-0.25-10.125
0.75-0.125

Questions

What are the solutions of 2x² + 9x - 8 = 0?

The solutions of 2x² + 9x - 8 = 0 are x = (-9 + √145)/4 or x = (-9 - √145)/4, that is x ≈ 0.76039864 and x ≈ -5.2603986.

Does 2x² + 9x - 8 = 0 factor?

Not over the integers, because the discriminant 145 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 9x - 8?

(-9/4, -145/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = -4.

Similar equations

    Solve 2x² + 9x - 8 = 0: x = (-9 + √145)/4 or x = (-9 - √145)/4 (steps) | WordGPT