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Solve 2x² + 3x - 5 = 0

The solutions of 2x² + 3x - 5 = 0 are x = 1 or x = -5/2.

x² +x += 0
2x2+3x−5=0⇒x=1, −522x^2 + 3x - 5 = 0 \Rightarrow x = 1,\ -\frac{5}{2}
Discriminant D = 49 · 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 + 3x - 5-5-4-3-2-1123-10-551015202530x = 1x = -2.5vertexy = 2x² + 3x - 5
Graph of y = 2x² + 3x - 5. The curve crosses the x-axis at x = 1 and x = -2.5; the vertex is at (-0.75, -6.125).

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 = 3, c = -5.

    2x2+3x−5=02x^2 + 3x - 5 = 0
  2. 2

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

    D=32−4⋅2⋅(−5)=9+40=49D = 3^2 - 4\cdot2\cdot(-5) = 9 + 40 = 49
  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=−3±494x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{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) + 3·(1) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 3x - 5 = 0\ \Rightarrow\ x = 1,\ -\frac{5}{2} \]
\[ 2x^2 + 3x - 5 = 0 \]
\[ D = 3^2 - 4\cdot2\cdot(-5) = 9 + 40 = 49 \]
\[ \left(x - 1\right)\left(2x + 5\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{4} \]
\[ x = 1\quad\text{or}\quad x = -\frac{5}{2} \]

The parabola y = 2x² + 3x - 5

xy
-3.7511.875
-2.751.875
-1.75-4.125
-0.75-6.125
0.25-4.125
1.251.875
2.2511.875

Questions

What are the solutions of 2x² + 3x - 5 = 0?

The solutions of 2x² + 3x - 5 = 0 are x = 1 or x = -5/2.

Does 2x² + 3x - 5 = 0 factor?

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

What is the vertex of y = 2x² + 3x - 5?

(-3/4, -49/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3/2, product = c/a = -5/2.

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