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Solve 2x² - 3x - 5 = 0

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

x² +x += 0
2x2−3x−5=0⇒x=52, −12x^2 - 3x - 5 = 0 \Rightarrow x = \frac{5}{2},\ -1
Discriminant D = 49 · x ≈ 2.5, -1

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

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Graph of y = 2x² - 3x - 5-3-2-112345-10-551015202530x = 2.5x = -1vertexy = 2x² - 3x - 5
Graph of y = 2x² - 3x - 5. The curve crosses the x-axis at x = 2.5 and x = -1; 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=(−3)2−4⋅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:

    (2x−5)(x+1)=0\left(2x - 5\right)\left(x + 1\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−5=0⇒x=52x+1=0⇒x=−12x - 5 = 0 \Rightarrow x = \frac{5}{2}\qquad x + 1 = 0 \Rightarrow x = -1
  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=52orx=−1x = \frac{5}{2}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 5/2 back in: a·x² + b·x + c = 2·(25/4) + (-3)·(5/2) + (-5) = 0 ✓

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

The parabola y = 2x² - 3x - 5

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

Questions

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

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

Does 2x² - 3x - 5 = 0 factor?

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