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Solve -2x² - 5x + 7 = 0

The solutions of -2x² - 5x + 7 = 0 are x = -7/2 or x = 1.

x² +x += 0
−2x2−5x+7=0⇒x=−72, 1-2x^2 - 5x + 7 = 0 \Rightarrow x = -\frac{7}{2},\ 1
Discriminant D = 81 · x ≈ -3.5, 1

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

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Graph of y = -2x² - 5x + 7-6-5-4-3-2-1123-30-20-1010x = -3.5x = 1vertexy = -2x² - 5x + 7
Graph of y = -2x² - 5x + 7. The curve crosses the x-axis at x = -3.5 and x = 1; the vertex is at (-1.25, 10.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = -5, c = 7.

    −2x2−5x+7=0-2x^2 - 5x + 7 = 0
  2. 2

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

    D=(−5)2−4⋅(−2)⋅7=25+56=81D = (-5)^2 - 4\cdot(-2)\cdot7 = 25 + 56 = 81
  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:

    −(2x+7)(x−1)=0-\left(2x + 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:

    2x+7=0⇒x=−72x−1=0⇒x=12x + 7 = 0 \Rightarrow x = -\frac{7}{2}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±81−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{81}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−72orx=1x = -\frac{7}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -7/2 back in: a·x² + b·x + c = -2·(49/4) + (-5)·(-7/2) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 5x + 7 = 0\ \Rightarrow\ x = -\frac{7}{2},\ 1 \]
\[ -2x^2 - 5x + 7 = 0 \]
\[ D = (-5)^2 - 4\cdot(-2)\cdot7 = 25 + 56 = 81 \]
\[ -\left(2x + 7\right)\left(x - 1\right) = 0 \]
\[ 2x + 7 = 0 \Rightarrow x = -\frac{7}{2}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{81}}{-4} \]
\[ x = -\frac{7}{2}\quad\text{or}\quad x = 1 \]

The parabola y = -2x² - 5x + 7

xy
-4.25-7.875
-3.252.125
-2.258.125
-1.2510.125
-0.258.125
0.752.125
1.75-7.875

Questions

What are the solutions of -2x² - 5x + 7 = 0?

The solutions of -2x² - 5x + 7 = 0 are x = -7/2 or x = 1.

Does -2x² - 5x + 7 = 0 factor?

Yes: -1(2x+7)(x-1) = 0.

What is the vertex of y = -2x² - 5x + 7?

(-5/4, 81/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = -7/2.

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