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Solve -x² + 8x - 7 = 0

The solutions of -x² + 8x - 7 = 0 are x = 1 or x = 7.

x² +x += 0
−x2+8x−7=0⇒x=1, 7-x^2 + 8x - 7 = 0 \Rightarrow x = 1,\ 7
Discriminant D = 36 · x ≈ 1, 7

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

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Graph of y = -x² + 8x - 72468-20-15-10-5510x = 1x = 7vertexy = -x² + 8x - 7
Graph of y = -x² + 8x - 7. The curve crosses the x-axis at x = 1 and x = 7; the vertex is at (4, 9).

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 = 8, c = -7.

    −x2+8x−7=0-x^2 + 8x - 7 = 0
  2. 2

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

    D=82−4⋅(−1)⋅(−7)=64−28=36D = 8^2 - 4\cdot(-1)\cdot(-7) = 64 - 28 = 36
  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−7)=0-\left(x - 1\right)\left(x - 7\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−7=0⇒x=7x - 1 = 0 \Rightarrow x = 1\qquad x - 7 = 0 \Rightarrow x = 7
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±36−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{36}}{-2}
  7. 7

    Simplify to get the solutions:

    x=1orx=7x = 1\quad\text{or}\quad x = 7
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -1·(1) + 8·(1) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x - 7 = 0\ \Rightarrow\ x = 1,\ 7 \]
\[ -x^2 + 8x - 7 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot(-7) = 64 - 28 = 36 \]
\[ -\left(x - 1\right)\left(x - 7\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 7 = 0 \Rightarrow x = 7 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{36}}{-2} \]
\[ x = 1\quad\text{or}\quad x = 7 \]

The parabola y = -x² + 8x - 7

xy
10
25
38
49
58
65
70

Questions

What are the solutions of -x² + 8x - 7 = 0?

The solutions of -x² + 8x - 7 = 0 are x = 1 or x = 7.

Does -x² + 8x - 7 = 0 factor?

Yes: -1(x-1)(x-7) = 0.

What is the vertex of y = -x² + 8x - 7?

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

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = 7.

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