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Solve 2x² + 7x + 5 = 0

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

x² +x += 0
2x2+7x+5=0⇒x=−1, −522x^2 + 7x + 5 = 0 \Rightarrow x = -1,\ -\frac{5}{2}
Discriminant D = 9 · x ≈ -1, -2.5

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

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Graph of y = 2x² + 7x + 5-5-4-3-2-115101520x = -1x = -2.5vertexy = 2x² + 7x + 5
Graph of y = 2x² + 7x + 5. The curve crosses the x-axis at x = -1 and x = -2.5; the vertex is at (-1.75, -1.125).

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 = 7, c = 5.

    2x2+7x+5=02x^2 + 7x + 5 = 0
  2. 2

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

    D=72−4⋅2⋅5=49−40=9D = 7^2 - 4\cdot2\cdot5 = 49 - 40 = 9
  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+5)=0\left(x + 1\right)\left(2x + 5\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+5=0⇒x=−52x + 1 = 0 \Rightarrow x = -1\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±94x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−52x = -1\quad\text{or}\quad x = -\frac{5}{2}
  8. 8

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

LaTeX source of the formulas on this page
\[ 2x^2 + 7x + 5 = 0\ \Rightarrow\ x = -1,\ -\frac{5}{2} \]
\[ 2x^2 + 7x + 5 = 0 \]
\[ D = 7^2 - 4\cdot2\cdot5 = 49 - 40 = 9 \]
\[ \left(x + 1\right)\left(2x + 5\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{9}}{4} \]
\[ x = -1\quad\text{or}\quad x = -\frac{5}{2} \]

The parabola y = 2x² + 7x + 5

xy
-4.7516.875
-3.756.875
-2.750.875
-1.75-1.125
-0.750.875
0.256.875
1.2516.875

Questions

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

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

Does 2x² + 7x + 5 = 0 factor?

Yes: (x+1)(2x+5) = 0.

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

(-7/4, -9/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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