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Solve 4x² - 2x - 6 = 0

The solutions of 4x² - 2x - 6 = 0 are x = 3/2 or x = -1.

x² +x += 0
4x2−2x−6=0⇒x=32, −14x^2 - 2x - 6 = 0 \Rightarrow x = \frac{3}{2},\ -1
Discriminant D = 100 · x ≈ 1.5, -1

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

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Graph of y = 4x² - 2x - 6-3-2-11234-101020304050x = 1.5x = -1vertexy = 4x² - 2x - 6
Graph of y = 4x² - 2x - 6. The curve crosses the x-axis at x = 1.5 and x = -1; the vertex is at (0.25, -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 = -2, c = -6.

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

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

    D=(−2)2−4⋅4⋅(−6)=4+96=100D = (-2)^2 - 4\cdot4\cdot(-6) = 4 + 96 = 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(2x−3)(x+1)=02\left(2x - 3\right)\left(x + 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:

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

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 4x^2 - 2x - 6 = 0\ \Rightarrow\ x = \frac{3}{2},\ -1 \]
\[ 4x^2 - 2x - 6 = 0 \]
\[ D = (-2)^2 - 4\cdot4\cdot(-6) = 4 + 96 = 100 \]
\[ 2\left(2x - 3\right)\left(x + 1\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{100}}{8} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = -1 \]

The parabola y = 4x² - 2x - 6

xy
-2.7529.75
-1.759.75
-0.75-2.25
0.25-6.25
1.25-2.25
2.259.75
3.2529.75

Questions

What are the solutions of 4x² - 2x - 6 = 0?

The solutions of 4x² - 2x - 6 = 0 are x = 3/2 or x = -1.

Does 4x² - 2x - 6 = 0 factor?

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

What is the vertex of y = 4x² - 2x - 6?

(1/4, -25/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/2, product = c/a = -3/2.

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