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Solve 4x² - 11x + 6 = 0

The solutions of 4x² - 11x + 6 = 0 are x = 2 or x = 3/4.

x² +x += 0
4x2−11x+6=0⇒x=2, 344x^2 - 11x + 6 = 0 \Rightarrow x = 2,\ \frac{3}{4}
Discriminant D = 25 · x ≈ 2, 0.75

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

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Graph of y = 4x² - 11x + 6-1123410203040x = 2x = 0.75vertexy = 4x² - 11x + 6
Graph of y = 4x² - 11x + 6. The curve crosses the x-axis at x = 2 and x = 0.75; the vertex is at (1.375, -1.5625).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -11, c = 6.

    4x2−11x+6=04x^2 - 11x + 6 = 0
  2. 2

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

    D=(−11)2−4⋅4⋅6=121−96=25D = (-11)^2 - 4\cdot4\cdot6 = 121 - 96 = 25
  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)(4x−3)=0\left(x - 2\right)\left(4x - 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=24x−3=0⇒x=34x - 2 = 0 \Rightarrow x = 2\qquad 4x - 3 = 0 \Rightarrow x = \frac{3}{4}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±258x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{25}}{8}
  7. 7

    Simplify to get the solutions:

    x=2orx=34x = 2\quad\text{or}\quad x = \frac{3}{4}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 4·(4) + (-11)·(2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 11x + 6 = 0\ \Rightarrow\ x = 2,\ \frac{3}{4} \]
\[ 4x^2 - 11x + 6 = 0 \]
\[ D = (-11)^2 - 4\cdot4\cdot6 = 121 - 96 = 25 \]
\[ \left(x - 2\right)\left(4x - 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 4x - 3 = 0 \Rightarrow x = \frac{3}{4} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{25}}{8} \]
\[ x = 2\quad\text{or}\quad x = \frac{3}{4} \]

The parabola y = 4x² - 11x + 6

xy
-1.62534.4375
-0.62514.4375
0.3752.4375
1.375-1.5625
2.3752.4375
3.37514.4375
4.37534.4375

Questions

What are the solutions of 4x² - 11x + 6 = 0?

The solutions of 4x² - 11x + 6 = 0 are x = 2 or x = 3/4.

Does 4x² - 11x + 6 = 0 factor?

Yes: (x-2)(4x-3) = 0.

What is the vertex of y = 4x² - 11x + 6?

(11/8, -25/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/4, product = c/a = 3/2.

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