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Solve -x² + 2x + 3 = 0

The solutions of -x² + 2x + 3 = 0 are x = -1 or x = 3.

x² +x += 0
−x2+2x+3=0⇒x=−1, 3-x^2 + 2x + 3 = 0 \Rightarrow x = -1,\ 3
Discriminant D = 16 · x ≈ -1, 3

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

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Graph of y = -x² + 2x + 3-3-2-112345-15-10-55x = -1x = 3vertexy = -x² + 2x + 3
Graph of y = -x² + 2x + 3. The curve crosses the x-axis at x = -1 and x = 3; the vertex is at (1, 4).

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 = 2, c = 3.

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

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

    D=22−4⋅(−1)⋅3=4+12=16D = 2^2 - 4\cdot(-1)\cdot3 = 4 + 12 = 16
  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+1)(x−3)=0-\left(x + 1\right)\left(x - 3\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+1=0⇒x=−1x−3=0⇒x=3x + 1 = 0 \Rightarrow x = -1\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−2±16−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{16}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=3x = -1\quad\text{or}\quad x = 3
  8. 8

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

LaTeX source of the formulas on this page
\[ -x^2 + 2x + 3 = 0\ \Rightarrow\ x = -1,\ 3 \]
\[ -x^2 + 2x + 3 = 0 \]
\[ D = 2^2 - 4\cdot(-1)\cdot3 = 4 + 12 = 16 \]
\[ -\left(x + 1\right)\left(x - 3\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{16}}{-2} \]
\[ x = -1\quad\text{or}\quad x = 3 \]

The parabola y = -x² + 2x + 3

xy
-2-5
-10
03
14
23
30
4-5

Questions

What are the solutions of -x² + 2x + 3 = 0?

The solutions of -x² + 2x + 3 = 0 are x = -1 or x = 3.

Does -x² + 2x + 3 = 0 factor?

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

What is the vertex of y = -x² + 2x + 3?

(1, 4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -3.

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