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Solve 5x² - 7 = 0

The solutions of 5x² - 7 = 0 are x = (√35)/5 or x = (-√35)/5, that is x ≈ 1.183216 and x ≈ -1.183216.

x² +x += 0
5x2−7=0⇒x=355, −3555x^2 - 7 = 0 \Rightarrow x = \frac{\sqrt{35}}{5},\ \frac{-\sqrt{35}}{5}
Discriminant D = 140 · x ≈ 1.18322, -1.18322

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

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Graph of y = 5x² - 7-3-2-1123-10102030405060x = 1.183216x = -1.183216vertexy = 5x² - 7
Graph of y = 5x² - 7. The curve crosses the x-axis at x = 1.183216 and x = -1.183216; the vertex is at (0, -7).

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 = -7.

    5x2−7=05x^2 - 7 = 0
  2. 2

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

    D=02−4⋅5⋅(−7)=0+140=140D = 0^2 - 4\cdot5\cdot(-7) = 0 + 140 = 140
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=0±14010x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{140}}{10}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=355orx=−355x = \frac{\sqrt{35}}{5}\quad\text{or}\quad x = \frac{-\sqrt{35}}{5}
  6. 6

    As decimals: x ≈ 1.183216 and x ≈ -1.183216.

  7. 7

    Check numerically: for x ≈ 1.183216 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 7 = 0\ \Rightarrow\ x = \frac{\sqrt{35}}{5},\ \frac{-\sqrt{35}}{5} \]
\[ 5x^2 - 7 = 0 \]
\[ D = 0^2 - 4\cdot5\cdot(-7) = 0 + 140 = 140 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{140}}{10} \]
\[ x = \frac{\sqrt{35}}{5}\quad\text{or}\quad x = \frac{-\sqrt{35}}{5} \]

The parabola y = 5x² - 7

xy
-338
-213
-1-2
0-7
1-2
213
338

Questions

What are the solutions of 5x² - 7 = 0?

The solutions of 5x² - 7 = 0 are x = (√35)/5 or x = (-√35)/5, that is x ≈ 1.183216 and x ≈ -1.183216.

Does 5x² - 7 = 0 factor?

Not over the integers, because the discriminant 140 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 5x² - 7?

(0, -7). 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 = -7/5.

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    Solve 5x² - 7 = 0: x = (√35)/5 or x = (-√35)/5 (steps) | WordGPT