WordGPT

Solve -x² - 6x + 7 = 0

The solutions of -x² - 6x + 7 = 0 are x = -7 or x = 1.

x² +x += 0
−x2−6x+7=0⇒x=−7, 1-x^2 - 6x + 7 = 0 \Rightarrow x = -7,\ 1
Discriminant D = 64 · x ≈ -7, 1

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

Explain it step by step in chatMake a short video of it
Graph of y = -x² - 6x + 7-8-6-4-22-20-101020x = -7x = 1vertexy = -x² - 6x + 7
Graph of y = -x² - 6x + 7. The curve crosses the x-axis at x = -7 and x = 1; the vertex is at (-3, 16).

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 = -6, c = 7.

    −x2−6x+7=0-x^2 - 6x + 7 = 0
  2. 2

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

    D=(−6)2−4⋅(−1)⋅7=36+28=64D = (-6)^2 - 4\cdot(-1)\cdot7 = 36 + 28 = 64
  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+7)(x−1)=0-\left(x + 7\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+7=0⇒x=−7x−1=0⇒x=1x + 7 = 0 \Rightarrow x = -7\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±64−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{64}}{-2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -7 back in: a·x² + b·x + c = -1·(49) + (-6)·(-7) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 6x + 7 = 0\ \Rightarrow\ x = -7,\ 1 \]
\[ -x^2 - 6x + 7 = 0 \]
\[ D = (-6)^2 - 4\cdot(-1)\cdot7 = 36 + 28 = 64 \]
\[ -\left(x + 7\right)\left(x - 1\right) = 0 \]
\[ x + 7 = 0 \Rightarrow x = -7\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{64}}{-2} \]
\[ x = -7\quad\text{or}\quad x = 1 \]

The parabola y = -x² - 6x + 7

xy
-67
-512
-415
-316
-215
-112
07

Questions

What are the solutions of -x² - 6x + 7 = 0?

The solutions of -x² - 6x + 7 = 0 are x = -7 or x = 1.

Does -x² - 6x + 7 = 0 factor?

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

What is the vertex of y = -x² - 6x + 7?

(-3, 16). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -6, product = c/a = -7.

Similar equations

    Solve -x² - 6x + 7 = 0: x = -7 or x = 1 (steps) | WordGPT