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Solve 4x² - 1 = 0

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

x² +x += 0
4x2−1=0⇒x=12, −124x^2 - 1 = 0 \Rightarrow x = \frac{1}{2},\ -\frac{1}{2}
Discriminant D = 16 · x ≈ 0.5, -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 - 1-3-2-1123-55101520253035x = 0.5x = -0.5vertexy = 4x² - 1
Graph of y = 4x² - 1. The curve crosses the x-axis at x = 0.5 and x = -0.5; the vertex is at (0, -1).

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 = 0, c = -1.

    4x2−1=04x^2 - 1 = 0
  2. 2

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

    D=02−4⋅4⋅(−1)=0+16=16D = 0^2 - 4\cdot4\cdot(-1) = 0 + 16 = 16
  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−1)(2x+1)=0\left(2x - 1\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:

    2x−1=0⇒x=122x+1=0⇒x=−122x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=0±168x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{16}}{8}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 1/2 back in: a·x² + b·x + c = 4·(1/4) + 0·(1/2) + (-1) = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 1 = 0\ \Rightarrow\ x = \frac{1}{2},\ -\frac{1}{2} \]
\[ 4x^2 - 1 = 0 \]
\[ D = 0^2 - 4\cdot4\cdot(-1) = 0 + 16 = 16 \]
\[ \left(2x - 1\right)\left(2x + 1\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{16}}{8} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = 4x² - 1

xy
-335
-215
-13
0-1
13
215
335

Questions

What are the solutions of 4x² - 1 = 0?

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

Does 4x² - 1 = 0 factor?

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

What is the vertex of y = 4x² - 1?

(0, -1). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -1/4.

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