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Solve 2x² - 7x + 6 = 0

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

x² +x += 0
2x2−7x+6=0⇒x=2, 322x^2 - 7x + 6 = 0 \Rightarrow x = 2,\ \frac{3}{2}
Discriminant D = 1 · x ≈ 2, 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 - 7x + 6-11234-2246810121416x = 2x = 1.5vertexy = 2x² - 7x + 6
Graph of y = 2x² - 7x + 6. The curve crosses the x-axis at x = 2 and x = 1.5; 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=02x^2 - 7x + 6 = 0
  2. 2

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

    D=(−7)2−4⋅2⋅6=49−48=1D = (-7)^2 - 4\cdot2\cdot6 = 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:

    (x−2)(2x−3)=0\left(x - 2\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−2=0⇒x=22x−3=0⇒x=32x - 2 = 0 \Rightarrow x = 2\qquad 2x - 3 = 0 \Rightarrow x = \frac{3}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±14x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{4}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 2x^2 - 7x + 6 = 0\ \Rightarrow\ x = 2,\ \frac{3}{2} \]
\[ 2x^2 - 7x + 6 = 0 \]
\[ D = (-7)^2 - 4\cdot2\cdot6 = 49 - 48 = 1 \]
\[ \left(x - 2\right)\left(2x - 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 2x - 3 = 0 \Rightarrow x = \frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{4} \]
\[ x = 2\quad\text{or}\quad x = \frac{3}{2} \]

The parabola y = 2x² - 7x + 6

xy
-1.2517.875
-0.257.875
0.751.875
1.75-0.125
2.751.875
3.757.875
4.7517.875

Questions

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

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

Does 2x² - 7x + 6 = 0 factor?

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

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

(7/4, -1/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/2, product = c/a = 3.

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