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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, 12-2x^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 + 3-5-4-3-2-1123-30-25-20-15-10-5510x = -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=0-2x^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\cdot(-2)\cdot3 = 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±49−4x = \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
\[ -2x^2 - 5x + 3 = 0\ \Rightarrow\ x = -3,\ \frac{1}{2} \]
\[ -2x^2 - 5x + 3 = 0 \]
\[ D = (-5)^2 - 4\cdot(-2)\cdot3 = 25 + 24 = 49 \]
\[ -\left(x + 3\right)\left(2x - 1\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{-4} \]
\[ x = -3\quad\text{or}\quad x = \frac{1}{2} \]

The parabola y = -2x² - 5x + 3

xy
-4.25-11.875
-3.25-1.875
-2.254.125
-1.256.125
-0.254.125
0.75-1.875
1.75-11.875

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

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

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

What are the sum and product of the roots?

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

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