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Solve 2x² + 5x - 3 = 0

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

x² +x += 0
2x2+5x−3=0⇒x=12, −32x^2 + 5x - 3 = 0 \Rightarrow x = \frac{1}{2},\ -3
Discriminant D = 49 · x ≈ 0.5, -3

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

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Graph of y = 2x² + 5x - 3-5-4-3-2-1123-10-551015202530x = 0.5x = -3vertexy = 2x² + 5x - 3
Graph of y = 2x² + 5x - 3. The curve crosses the x-axis at x = 0.5 and x = -3; 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=52−4⋅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:

    (2x−1)(x+3)=0\left(2x - 1\right)\left(x + 3\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+3=0⇒x=−32x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x + 3 = 0 \Rightarrow x = -3
  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=12orx=−3x = \frac{1}{2}\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 1/2 back in: a·x² + b·x + c = 2·(1/4) + 5·(1/2) + (-3) = 0 ✓

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

The parabola y = 2x² + 5x - 3

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

Questions

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

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

Does 2x² + 5x - 3 = 0 factor?

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