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Solve -2x² + 7x - 6 = 0

The solutions of -2x² + 7x - 6 = 0 are x = 3/2 or x = 2.

x² +x += 0
−2x2+7x−6=0⇒x=32, 2-2x^2 + 7x - 6 = 0 \Rightarrow x = \frac{3}{2},\ 2
Discriminant D = 1 · x ≈ 1.5, 2

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

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Graph of y = -2x² + 7x - 6-11234-16-14-12-10-8-6-4-22x = 1.5x = 2vertexy = -2x² + 7x - 6
Graph of y = -2x² + 7x - 6. The curve crosses the x-axis at x = 1.5 and x = 2; the vertex is at (1.75, 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 = 7, c = -6.

    −2x2+7x−6=0-2x^2 + 7x - 6 = 0
  2. 2

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

    D=72−4⋅(−2)⋅(−6)=49−48=1D = 7^2 - 4\cdot(-2)\cdot(-6) = 49 - 48 = 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:

    −(2x−3)(x−2)=0-\left(2x - 3\right)\left(x - 2\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−2=0⇒x=22x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = 3/2 back in: a·x² + b·x + c = -2·(9/4) + 7·(3/2) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 7x - 6 = 0\ \Rightarrow\ x = \frac{3}{2},\ 2 \]
\[ -2x^2 + 7x - 6 = 0 \]
\[ D = 7^2 - 4\cdot(-2)\cdot(-6) = 49 - 48 = 1 \]
\[ -\left(2x - 3\right)\left(x - 2\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{-4} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = 2 \]

The parabola y = -2x² + 7x - 6

xy
-1.25-17.875
-0.25-7.875
0.75-1.875
1.750.125
2.75-1.875
3.75-7.875
4.75-17.875

Questions

What are the solutions of -2x² + 7x - 6 = 0?

The solutions of -2x² + 7x - 6 = 0 are x = 3/2 or x = 2.

Does -2x² + 7x - 6 = 0 factor?

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

What is the vertex of y = -2x² + 7x - 6?

(7/4, 1/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 7/2, product = c/a = 3.

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