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Solve 4x² - 6x - 4 = 0

The solutions of 4x² - 6x - 4 = 0 are x = 2 or x = -1/2.

x² +x += 0
4x2−6x−4=0⇒x=2, −124x^2 - 6x - 4 = 0 \Rightarrow x = 2,\ -\frac{1}{2}
Discriminant D = 100 · x ≈ 2, -0.5

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

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Graph of y = 4x² - 6x - 4-3-2-11234-101020304050x = 2x = -0.5vertexy = 4x² - 6x - 4
Graph of y = 4x² - 6x - 4. The curve crosses the x-axis at x = 2 and x = -0.5; the vertex is at (0.75, -6.25).

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 = -6, c = -4.

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

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

    D=(−6)2−4⋅4⋅(−4)=36+64=100D = (-6)^2 - 4\cdot4\cdot(-4) = 36 + 64 = 100
  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:

    2(x−2)(2x+1)=02\left(x - 2\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−2=0⇒x=22x+1=0⇒x=−12x - 2 = 0 \Rightarrow x = 2\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±1008x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{100}}{8}
  7. 7

    Simplify to get the solutions:

    x=2orx=−12x = 2\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 - 6x - 4 = 0\ \Rightarrow\ x = 2,\ -\frac{1}{2} \]
\[ 4x^2 - 6x - 4 = 0 \]
\[ D = (-6)^2 - 4\cdot4\cdot(-4) = 36 + 64 = 100 \]
\[ 2\left(x - 2\right)\left(2x + 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{100}}{8} \]
\[ x = 2\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = 4x² - 6x - 4

xy
-2.2529.75
-1.259.75
-0.25-2.25
0.75-6.25
1.75-2.25
2.759.75
3.7529.75

Questions

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

The solutions of 4x² - 6x - 4 = 0 are x = 2 or x = -1/2.

Does 4x² - 6x - 4 = 0 factor?

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

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

(3/4, -25/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = -1.

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