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Solve 2x² + 9x - 5 = 0

The solutions of 2x² + 9x - 5 = 0 are x = 1/2 or x = -5.

x² +x += 0
2x2+9x−5=0⇒x=12, −52x^2 + 9x - 5 = 0 \Rightarrow x = \frac{1}{2},\ -5
Discriminant D = 121 · x ≈ 0.5, -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 - 5-6-4-22-20-1010203040x = 0.5x = -5vertexy = 2x² + 9x - 5
Graph of y = 2x² + 9x - 5. The curve crosses the x-axis at x = 0.5 and x = -5; the vertex is at (-2.25, -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 = 9, c = -5.

    2x2+9x−5=02x^2 + 9x - 5 = 0
  2. 2

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

    D=92−4⋅2⋅(−5)=81+40=121D = 9^2 - 4\cdot2\cdot(-5) = 81 + 40 = 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:

    (2x−1)(x+5)=0\left(2x - 1\right)\left(x + 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:

    2x−1=0⇒x=12x+5=0⇒x=−52x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 5 = 0 \Rightarrow x = -5
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±1214x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{121}}{4}
  7. 7

    Simplify to get the solutions:

    x=12orx=−5x = \frac{1}{2}\quad\text{or}\quad x = -5
  8. 8

    Check by putting x = 1/2 back in: a·x² + b·x + c = 2·(1/4) + 9·(1/2) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x - 5 = 0\ \Rightarrow\ x = \frac{1}{2},\ -5 \]
\[ 2x^2 + 9x - 5 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot(-5) = 81 + 40 = 121 \]
\[ \left(2x - 1\right)\left(x + 5\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 5 = 0 \Rightarrow x = -5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{121}}{4} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = -5 \]

The parabola y = 2x² + 9x - 5

xy
-5.252.875
-4.25-7.125
-3.25-13.125
-2.25-15.125
-1.25-13.125
-0.25-7.125
0.752.875

Questions

What are the solutions of 2x² + 9x - 5 = 0?

The solutions of 2x² + 9x - 5 = 0 are x = 1/2 or x = -5.

Does 2x² + 9x - 5 = 0 factor?

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

What is the vertex of y = 2x² + 9x - 5?

(-9/4, -121/8). 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 = -5/2.

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