WordGPT

Solve -3x² + 11x - 10 = 0

The solutions of -3x² + 11x - 10 = 0 are x = 5/3 or x = 2.

x² +x += 0
−3x2+11x−10=0⇒x=53, 2-3x^2 + 11x - 10 = 0 \Rightarrow x = \frac{5}{3},\ 2
Discriminant D = 1 · x ≈ 1.66667, 2

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 + 11x - 101234-20-15-10-5x = 1.6666667x = 2vertexy = -3x² + 11x - 10
Graph of y = -3x² + 11x - 10. The curve crosses the x-axis at x = 1.6666667 and x = 2; the vertex is at (1.8333333, 0.083333333).

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 = 11, c = -10.

    −3x2+11x−10=0-3x^2 + 11x - 10 = 0
  2. 2

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

    D=112−4⋅(−3)⋅(−10)=121−120=1D = 11^2 - 4\cdot(-3)\cdot(-10) = 121 - 120 = 1
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(3x−5)(x−2)=0-\left(3x - 5\right)\left(x - 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    3x−5=0⇒x=53x−2=0⇒x=23x - 5 = 0 \Rightarrow x = \frac{5}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=53orx=2x = \frac{5}{3}\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = 5/3 back in: a·x² + b·x + c = -3·(25/9) + 11·(5/3) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 11x - 10 = 0\ \Rightarrow\ x = \frac{5}{3},\ 2 \]
\[ -3x^2 + 11x - 10 = 0 \]
\[ D = 11^2 - 4\cdot(-3)\cdot(-10) = 121 - 120 = 1 \]
\[ -\left(3x - 5\right)\left(x - 2\right) = 0 \]
\[ 3x - 5 = 0 \Rightarrow x = \frac{5}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{1}}{-6} \]
\[ x = \frac{5}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 11x - 10

xy
-1.1667-26.917267
-0.1667-11.917067
0.8333-2.9168667
1.83330.08333333
2.8333-2.9164667
3.8333-11.916267
4.8333-26.916067

Questions

What are the solutions of -3x² + 11x - 10 = 0?

The solutions of -3x² + 11x - 10 = 0 are x = 5/3 or x = 2.

Does -3x² + 11x - 10 = 0 factor?

Yes: -1(3x-5)(x-2) = 0.

What is the vertex of y = -3x² + 11x - 10?

(11/6, 1/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 11/3, product = c/a = 10/3.

Similar equations

    Solve -3x² + 11x - 10 = 0: x = 5/3 or x = 2 (steps) | WordGPT