WordGPT

Solve 3x² + 8x - 10 = 0

The solutions of 3x² + 8x - 10 = 0 are x = (-4 + √46)/3 or x = (-4 - √46)/3, that is x ≈ 0.92744333 and x ≈ -3.59411.

x² +x += 0
3x2+8x−10=0⇒x=−4+463, −4−4633x^2 + 8x - 10 = 0 \Rightarrow x = \frac{-4 + \sqrt{46}}{3},\ \frac{-4 - \sqrt{46}}{3}
Discriminant D = 184 · x ≈ 0.927443, -3.59411

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 = 3x² + 8x - 10-6-5-4-3-2-1123-20-101020304050x = 0.92744333x = -3.59411vertexy = 3x² + 8x - 10
Graph of y = 3x² + 8x - 10. The curve crosses the x-axis at x = 0.92744333 and x = -3.59411; the vertex is at (-1.3333333, -15.333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = 8, c = -10.

    3x2+8x−10=03x^2 + 8x - 10 = 0
  2. 2

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

    D=82−4⋅3⋅(−10)=64+120=184D = 8^2 - 4\cdot3\cdot(-10) = 64 + 120 = 184
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±1846x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{184}}{6}
  5. 5

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

    x=−4+463orx=−4−463x = \frac{-4 + \sqrt{46}}{3}\quad\text{or}\quad x = \frac{-4 - \sqrt{46}}{3}
  6. 6

    As decimals: x ≈ 0.92744333 and x ≈ -3.59411.

  7. 7

    Check numerically: for x ≈ 0.92744333 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 8x - 10 = 0\ \Rightarrow\ x = \frac{-4 + \sqrt{46}}{3},\ \frac{-4 - \sqrt{46}}{3} \]
\[ 3x^2 + 8x - 10 = 0 \]
\[ D = 8^2 - 4\cdot3\cdot(-10) = 64 + 120 = 184 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{184}}{6} \]
\[ x = \frac{-4 + \sqrt{46}}{3}\quad\text{or}\quad x = \frac{-4 - \sqrt{46}}{3} \]

The parabola y = 3x² + 8x - 10

xy
-4.333311.666067
-3.3333-3.3337333
-2.3333-12.333533
-1.3333-15.333333
-0.3333-12.333133
0.6667-3.3329333
1.666711.667267

Questions

What are the solutions of 3x² + 8x - 10 = 0?

The solutions of 3x² + 8x - 10 = 0 are x = (-4 + √46)/3 or x = (-4 - √46)/3, that is x ≈ 0.92744333 and x ≈ -3.59411.

Does 3x² + 8x - 10 = 0 factor?

Not over the integers, because the discriminant 184 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² + 8x - 10?

(-4/3, -46/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = -10/3.

Similar equations

    Solve 3x² + 8x - 10 = 0: x = (-4 + √46)/3 or x = (-4 - √46)/3 (steps) | WordGPT