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Solve -x² + 5x - 4 = 0

The solutions of -x² + 5x - 4 = 0 are x = 1 or x = 4.

x² +x += 0
−x2+5x−4=0⇒x=1, 4-x^2 + 5x - 4 = 0 \Rightarrow x = 1,\ 4
Discriminant D = 9 · x ≈ 1, 4

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

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Graph of y = -x² + 5x - 4-1123456-14-12-10-8-6-4-224x = 1x = 4vertexy = -x² + 5x - 4
Graph of y = -x² + 5x - 4. The curve crosses the x-axis at x = 1 and x = 4; the vertex is at (2.5, 2.25).

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 = 5, c = -4.

    −x2+5x−4=0-x^2 + 5x - 4 = 0
  2. 2

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

    D=52−4⋅(−1)⋅(−4)=25−16=9D = 5^2 - 4\cdot(-1)\cdot(-4) = 25 - 16 = 9
  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−4)=0-\left(x - 1\right)\left(x - 4\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−4=0⇒x=4x - 1 = 0 \Rightarrow x = 1\qquad x - 4 = 0 \Rightarrow x = 4
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±9−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{-2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 1 back in: a·x² + b·x + c = -1·(1) + 5·(1) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 5x - 4 = 0\ \Rightarrow\ x = 1,\ 4 \]
\[ -x^2 + 5x - 4 = 0 \]
\[ D = 5^2 - 4\cdot(-1)\cdot(-4) = 25 - 16 = 9 \]
\[ -\left(x - 1\right)\left(x - 4\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 4 = 0 \Rightarrow x = 4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{-2} \]
\[ x = 1\quad\text{or}\quad x = 4 \]

The parabola y = -x² + 5x - 4

xy
-0.5-6.75
0.5-1.75
1.51.25
2.52.25
3.51.25
4.5-1.75
5.5-6.75

Questions

What are the solutions of -x² + 5x - 4 = 0?

The solutions of -x² + 5x - 4 = 0 are x = 1 or x = 4.

Does -x² + 5x - 4 = 0 factor?

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

What is the vertex of y = -x² + 5x - 4?

(5/2, 9/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 4.

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