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Solve 2x² - x - 3 = 0

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

x² +x += 0
2x2−x−3=0⇒x=32, −12x^2 - x - 3 = 0 \Rightarrow x = \frac{3}{2},\ -1
Discriminant D = 25 · x ≈ 1.5, -1

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

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Graph of y = 2x² - x - 3-3-2-11234-5510152025x = 1.5x = -1vertexy = 2x² - x - 3
Graph of y = 2x² - x - 3. The curve crosses the x-axis at x = 1.5 and x = -1; the vertex is at (0.25, -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 = -1, c = -3.

    2x2−x−3=02x^2 - x - 3 = 0
  2. 2

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

    D=(−1)2−4⋅2⋅(−3)=1+24=25D = (-1)^2 - 4\cdot2\cdot(-3) = 1 + 24 = 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:

    (2x−3)(x+1)=0\left(2x - 3\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−3=0⇒x=32x+1=0⇒x=−12x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±254x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{4}
  7. 7

    Simplify to get the solutions:

    x=32orx=−1x = \frac{3}{2}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 3/2 back in: a·x² + b·x + c = 2·(9/4) + (-1)·(3/2) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - x - 3 = 0\ \Rightarrow\ x = \frac{3}{2},\ -1 \]
\[ 2x^2 - x - 3 = 0 \]
\[ D = (-1)^2 - 4\cdot2\cdot(-3) = 1 + 24 = 25 \]
\[ \left(2x - 3\right)\left(x + 1\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{4} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = -1 \]

The parabola y = 2x² - x - 3

xy
-2.7514.875
-1.754.875
-0.75-1.125
0.25-3.125
1.25-1.125
2.254.875
3.2514.875

Questions

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

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

Does 2x² - x - 3 = 0 factor?

Yes: (2x-3)(x+1) = 0.

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

(1/4, -25/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/2, product = c/a = -3/2.

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