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Solve 5x² - 5 = 0

The solutions of 5x² - 5 = 0 are x = 1 or x = -1.

x² +x += 0
5x2−5=0⇒x=1, −15x^2 - 5 = 0 \Rightarrow x = 1,\ -1
Discriminant D = 100 · x ≈ 1, -1

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

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Graph of y = 5x² - 5-3-2-1123-10102030405060x = 1x = -1vertexy = 5x² - 5
Graph of y = 5x² - 5. The curve crosses the x-axis at x = 1 and x = -1; the vertex is at (0, -5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 0, c = -5.

    5x2−5=05x^2 - 5 = 0
  2. 2

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

    D=02−4⋅5⋅(−5)=0+100=100D = 0^2 - 4\cdot5\cdot(-5) = 0 + 100 = 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:

    5(x−1)(x+1)=05\left(x - 1\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:

    x−1=0⇒x=1x+1=0⇒x=−1x - 1 = 0 \Rightarrow x = 1\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=0±10010x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{100}}{10}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 1 back in: a·x² + b·x + c = 5·(1) + 0·(1) + (-5) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 5 = 0\ \Rightarrow\ x = 1,\ -1 \]
\[ 5x^2 - 5 = 0 \]
\[ D = 0^2 - 4\cdot5\cdot(-5) = 0 + 100 = 100 \]
\[ 5\left(x - 1\right)\left(x + 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{100}}{10} \]
\[ x = 1\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - 5

xy
-340
-215
-10
0-5
10
215
340

Questions

What are the solutions of 5x² - 5 = 0?

The solutions of 5x² - 5 = 0 are x = 1 or x = -1.

Does 5x² - 5 = 0 factor?

Yes: 5(x-1)(x+1) = 0.

What is the vertex of y = 5x² - 5?

(0, -5). 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.

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