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Solve -2x² + 5x - 3 = 0

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

x² +x += 0
−2x2+5x−3=0⇒x=1, 32-2x^2 + 5x - 3 = 0 \Rightarrow x = 1,\ \frac{3}{2}
Discriminant D = 1 · x ≈ 1, 1.5

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

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Graph of y = -2x² + 5x - 3-11234-16-14-12-10-8-6-4-22x = 1x = 1.5vertexy = -2x² + 5x - 3
Graph of y = -2x² + 5x - 3. The curve crosses the x-axis at x = 1 and x = 1.5; the vertex is at (1.25, 0.125).

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=52−4⋅(−2)⋅(−3)=25−24=1D = 5^2 - 4\cdot(-2)\cdot(-3) = 25 - 24 = 1
  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−1)(2x−3)=0-\left(x - 1\right)\left(2x - 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±1−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{-4}
  7. 7

    Simplify to get the solutions:

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

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

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

The parabola y = -2x² + 5x - 3

xy
-1.75-17.875
-0.75-7.875
0.25-1.875
1.250.125
2.25-1.875
3.25-7.875
4.25-17.875

Questions

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

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

Does -2x² + 5x - 3 = 0 factor?

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

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

(5/4, 1/8). 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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