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Solve -x² + 3x + 10 = 0

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

x² +x += 0
−x2+3x+10=0⇒x=−2, 5-x^2 + 3x + 10 = 0 \Rightarrow x = -2,\ 5
Discriminant D = 49 · x ≈ -2, 5

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

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Graph of y = -x² + 3x + 10-4-2246-25-20-15-10-551015x = -2x = 5vertexy = -x² + 3x + 10
Graph of y = -x² + 3x + 10. The curve crosses the x-axis at x = -2 and x = 5; the vertex is at (1.5, 12.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 3, c = 10.

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

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

    D=32−4⋅(−1)⋅10=9+40=49D = 3^2 - 4\cdot(-1)\cdot10 = 9 + 40 = 49
  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:

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

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

    x+2=0⇒x=−2x−5=0⇒x=5x + 2 = 0 \Rightarrow x = -2\qquad x - 5 = 0 \Rightarrow x = 5
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±49−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−2orx=5x = -2\quad\text{or}\quad x = 5
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = -1·(4) + 3·(-2) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 3x + 10 = 0\ \Rightarrow\ x = -2,\ 5 \]
\[ -x^2 + 3x + 10 = 0 \]
\[ D = 3^2 - 4\cdot(-1)\cdot10 = 9 + 40 = 49 \]
\[ -\left(x + 2\right)\left(x - 5\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x - 5 = 0 \Rightarrow x = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{-2} \]
\[ x = -2\quad\text{or}\quad x = 5 \]

The parabola y = -x² + 3x + 10

xy
-1.53.25
-0.58.25
0.511.25
1.512.25
2.511.25
3.58.25
4.53.25

Questions

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

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

Does -x² + 3x + 10 = 0 factor?

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

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

(3/2, 49/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -10.

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