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Solve 2x² - 5x - 3 = 0

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

x² +x += 0
2x2−5x−3=0⇒x=3, −122x^2 - 5x - 3 = 0 \Rightarrow x = 3,\ -\frac{1}{2}
Discriminant D = 49 · x ≈ 3, -0.5

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

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Graph of y = 2x² - 5x - 3-3-2-112345-10-551015202530x = 3x = -0.5vertexy = 2x² - 5x - 3
Graph of y = 2x² - 5x - 3. The curve crosses the x-axis at x = 3 and x = -0.5; the vertex is at (1.25, -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 = -5, c = -3.

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

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

    D=(−5)2−4⋅2⋅(−3)=25+24=49D = (-5)^2 - 4\cdot2\cdot(-3) = 25 + 24 = 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−3)(2x+1)=0\left(x - 3\right)\left(2x + 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±494x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{4}
  7. 7

    Simplify to get the solutions:

    x=3orx=−12x = 3\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 2·(9) + (-5)·(3) + (-3) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 5x - 3 = 0\ \Rightarrow\ x = 3,\ -\frac{1}{2} \]
\[ 2x^2 - 5x - 3 = 0 \]
\[ D = (-5)^2 - 4\cdot2\cdot(-3) = 25 + 24 = 49 \]
\[ \left(x - 3\right)\left(2x + 1\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{4} \]
\[ x = 3\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = 2x² - 5x - 3

xy
-1.7511.875
-0.751.875
0.25-4.125
1.25-6.125
2.25-4.125
3.251.875
4.2511.875

Questions

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

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

Does 2x² - 5x - 3 = 0 factor?

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

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

(5/4, -49/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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