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Solve 2x² + 5x - 7 = 0

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

x² +x += 0
2x2+5x−7=0⇒x=1, −722x^2 + 5x - 7 = 0 \Rightarrow x = 1,\ -\frac{7}{2}
Discriminant D = 81 · x ≈ 1, -3.5

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

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Graph of y = 2x² + 5x - 7-6-5-4-3-2-1123-10102030x = 1x = -3.5vertexy = 2x² + 5x - 7
Graph of y = 2x² + 5x - 7. The curve crosses the x-axis at x = 1 and x = -3.5; 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=52−4⋅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:

    (x−1)(2x+7)=0\left(x - 1\right)\left(2x + 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=12x+7=0⇒x=−72x - 1 = 0 \Rightarrow x = 1\qquad 2x + 7 = 0 \Rightarrow x = -\frac{7}{2}
  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=1orx=−72x = 1\quad\text{or}\quad x = -\frac{7}{2}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 2·(1) + 5·(1) + (-7) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 5x - 7 = 0\ \Rightarrow\ x = 1,\ -\frac{7}{2} \]
\[ 2x^2 + 5x - 7 = 0 \]
\[ D = 5^2 - 4\cdot2\cdot(-7) = 25 + 56 = 81 \]
\[ \left(x - 1\right)\left(2x + 7\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x + 7 = 0 \Rightarrow x = -\frac{7}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{81}}{4} \]
\[ x = 1\quad\text{or}\quad x = -\frac{7}{2} \]

The parabola y = 2x² + 5x - 7

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

Questions

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

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

Does 2x² + 5x - 7 = 0 factor?

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