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Solve -3x² + x + 10 = 0

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

x² +x += 0
−3x2+x+10=0⇒x=−53, 2-3x^2 + x + 10 = 0 \Rightarrow x = -\frac{5}{3},\ 2
Discriminant D = 121 · x ≈ -1.66667, 2

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

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Graph of y = -3x² + x + 10-4-3-2-11234-50-40-30-20-1010x = -1.6666667x = 2vertexy = -3x² + x + 10
Graph of y = -3x² + x + 10. The curve crosses the x-axis at x = -1.6666667 and x = 2; the vertex is at (0.16666667, 10.083333).

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 = 1, c = 10.

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

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

    D=12−4⋅(−3)⋅10=1+120=121D = 1^2 - 4\cdot(-3)\cdot10 = 1 + 120 = 121
  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=−1±121−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{121}}{-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) + 1·(-5/3) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + x + 10 = 0\ \Rightarrow\ x = -\frac{5}{3},\ 2 \]
\[ -3x^2 + x + 10 = 0 \]
\[ D = 1^2 - 4\cdot(-3)\cdot10 = 1 + 120 = 121 \]
\[ -\left(3x + 5\right)\left(x - 2\right) = 0 \]
\[ 3x + 5 = 0 \Rightarrow x = -\frac{5}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{121}}{-6} \]
\[ x = -\frac{5}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + x + 10

xy
-2.8333-16.916067
-1.8333-1.9162667
-0.83337.0835333
0.166710.083333
1.16677.0831333
2.1667-1.9170667
3.1667-16.917267

Questions

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

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

Does -3x² + x + 10 = 0 factor?

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

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

(1/6, 121/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1/3, product = c/a = -10/3.

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