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Solve 2x² + 9x + 7 = 0

The solutions of 2x² + 9x + 7 = 0 are x = -1 or x = -7/2.

x² +x += 0
2x2+9x+7=0⇒x=−1, −722x^2 + 9x + 7 = 0 \Rightarrow x = -1,\ -\frac{7}{2}
Discriminant D = 25 · 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 + 9x + 7-6-5-4-3-2-11-5510152025x = -1x = -3.5vertexy = 2x² + 9x + 7
Graph of y = 2x² + 9x + 7. The curve crosses the x-axis at x = -1 and x = -3.5; the vertex is at (-2.25, -3.125).

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 = 9, c = 7.

    2x2+9x+7=02x^2 + 9x + 7 = 0
  2. 2

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

    D=92−4⋅2⋅7=81−56=25D = 9^2 - 4\cdot2\cdot7 = 81 - 56 = 25
  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=−9±254x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{25}}{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) + 9·(-1) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x + 7 = 0\ \Rightarrow\ x = -1,\ -\frac{7}{2} \]
\[ 2x^2 + 9x + 7 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot7 = 81 - 56 = 25 \]
\[ \left(x + 1\right)\left(2x + 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x + 7 = 0 \Rightarrow x = -\frac{7}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{25}}{4} \]
\[ x = -1\quad\text{or}\quad x = -\frac{7}{2} \]

The parabola y = 2x² + 9x + 7

xy
-5.2514.875
-4.254.875
-3.25-1.125
-2.25-3.125
-1.25-1.125
-0.254.875
0.7514.875

Questions

What are the solutions of 2x² + 9x + 7 = 0?

The solutions of 2x² + 9x + 7 = 0 are x = -1 or x = -7/2.

Does 2x² + 9x + 7 = 0 factor?

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

What is the vertex of y = 2x² + 9x + 7?

(-9/4, -25/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = 7/2.

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