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Solve -x² - 3x + 4 = 0

The solutions of -x² - 3x + 4 = 0 are x = -4 or x = 1.

x² +x += 0
−x2−3x+4=0⇒x=−4, 1-x^2 - 3x + 4 = 0 \Rightarrow x = -4,\ 1
Discriminant D = 25 · x ≈ -4, 1

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

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Graph of y = -x² - 3x + 4-6-5-4-3-2-1123-20-15-10-55x = -4x = 1vertexy = -x² - 3x + 4
Graph of y = -x² - 3x + 4. The curve crosses the x-axis at x = -4 and x = 1; the vertex is at (-1.5, 6.25).

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 = 4.

    −x2−3x+4=0-x^2 - 3x + 4 = 0
  2. 2

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

    D=(−3)2−4⋅(−1)⋅4=9+16=25D = (-3)^2 - 4\cdot(-1)\cdot4 = 9 + 16 = 25
  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+4)(x−1)=0-\left(x + 4\right)\left(x - 1\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+4=0⇒x=−4x−1=0⇒x=1x + 4 = 0 \Rightarrow x = -4\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±25−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{25}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−4orx=1x = -4\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -4 back in: a·x² + b·x + c = -1·(16) + (-3)·(-4) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 3x + 4 = 0\ \Rightarrow\ x = -4,\ 1 \]
\[ -x^2 - 3x + 4 = 0 \]
\[ D = (-3)^2 - 4\cdot(-1)\cdot4 = 9 + 16 = 25 \]
\[ -\left(x + 4\right)\left(x - 1\right) = 0 \]
\[ x + 4 = 0 \Rightarrow x = -4\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{25}}{-2} \]
\[ x = -4\quad\text{or}\quad x = 1 \]

The parabola y = -x² - 3x + 4

xy
-4.5-2.75
-3.52.25
-2.55.25
-1.56.25
-0.55.25
0.52.25
1.5-2.75

Questions

What are the solutions of -x² - 3x + 4 = 0?

The solutions of -x² - 3x + 4 = 0 are x = -4 or x = 1.

Does -x² - 3x + 4 = 0 factor?

Yes: -1(x+4)(x-1) = 0.

What is the vertex of y = -x² - 3x + 4?

(-3/2, 25/4). 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 = -4.

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