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Solve 2x² + 7x - 9 = 0

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

x² +x += 0
2x2+7x−9=0⇒x=1, −922x^2 + 7x - 9 = 0 \Rightarrow x = 1,\ -\frac{9}{2}
Discriminant D = 121 · x ≈ 1, -4.5

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

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Graph of y = 2x² + 7x - 9-6-4-22-20-1010203040x = 1x = -4.5vertexy = 2x² + 7x - 9
Graph of y = 2x² + 7x - 9. The curve crosses the x-axis at x = 1 and x = -4.5; the vertex is at (-1.75, -15.125).

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 = -9.

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

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

    D=72−4⋅2⋅(−9)=49+72=121D = 7^2 - 4\cdot2\cdot(-9) = 49 + 72 = 121
  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+9)=0\left(x - 1\right)\left(2x + 9\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+9=0⇒x=−92x - 1 = 0 \Rightarrow x = 1\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±1214x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{121}}{4}
  7. 7

    Simplify to get the solutions:

    x=1orx=−92x = 1\quad\text{or}\quad x = -\frac{9}{2}
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 2·(1) + 7·(1) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 7x - 9 = 0\ \Rightarrow\ x = 1,\ -\frac{9}{2} \]
\[ 2x^2 + 7x - 9 = 0 \]
\[ D = 7^2 - 4\cdot2\cdot(-9) = 49 + 72 = 121 \]
\[ \left(x - 1\right)\left(2x + 9\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x + 9 = 0 \Rightarrow x = -\frac{9}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{121}}{4} \]
\[ x = 1\quad\text{or}\quad x = -\frac{9}{2} \]

The parabola y = 2x² + 7x - 9

xy
-4.752.875
-3.75-7.125
-2.75-13.125
-1.75-15.125
-0.75-13.125
0.25-7.125
1.252.875

Questions

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

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

Does 2x² + 7x - 9 = 0 factor?

Yes: (x-1)(2x+9) = 0.

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

(-7/4, -121/8). 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 = -9/2.

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