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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, −12x^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 - 7-3-2-1123456-10102030x = 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=02x^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\cdot2\cdot(-7) = 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±814x = \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
\[ 2x^2 - 5x - 7 = 0\ \Rightarrow\ x = \frac{7}{2},\ -1 \]
\[ 2x^2 - 5x - 7 = 0 \]
\[ D = (-5)^2 - 4\cdot2\cdot(-7) = 25 + 56 = 81 \]
\[ \left(2x - 7\right)\left(x + 1\right) = 0 \]
\[ 2x - 7 = 0 \Rightarrow x = \frac{7}{2}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{81}}{4} \]
\[ x = \frac{7}{2}\quad\text{or}\quad x = -1 \]

The parabola y = 2x² - 5x - 7

xy
-1.757.875
-0.75-2.125
0.25-8.125
1.25-10.125
2.25-8.125
3.25-2.125
4.257.875

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

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

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

What are the sum and product of the roots?

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

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