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Solve x² - 7x - 5 = 0

The solutions of x² - 7x - 5 = 0 are x = (7 + √69)/2 or x = (7 - √69)/2, that is x ≈ 7.6533119 and x ≈ -0.65331193.

x² +x += 0
x2−7x−5=0⇒x=7+692, 7−692x^2 - 7x - 5 = 0 \Rightarrow x = \frac{7 + \sqrt{69}}{2},\ \frac{7 - \sqrt{69}}{2}
Discriminant D = 69 · x ≈ 7.65331, -0.653312

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

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Graph of y = x² - 7x - 5-2246810-20-10102030x = 7.6533119x = -0.65331193vertexy = x² - 7x - 5
Graph of y = x² - 7x - 5. The curve crosses the x-axis at x = 7.6533119 and x = -0.65331193; the vertex is at (3.5, -17.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -7, c = -5.

    x2−7x−5=0x^2 - 7x - 5 = 0
  2. 2

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

    D=(−7)2−4⋅1⋅(−5)=49+20=69D = (-7)^2 - 4\cdot1\cdot(-5) = 49 + 20 = 69
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=7±692x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{69}}{2}
  5. 5

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

    x=7+692orx=7−692x = \frac{7 + \sqrt{69}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{69}}{2}
  6. 6

    As decimals: x ≈ 7.6533119 and x ≈ -0.65331193.

  7. 7

    Check numerically: for x ≈ 7.6533119 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 7x - 5 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{69}}{2},\ \frac{7 - \sqrt{69}}{2} \]
\[ x^2 - 7x - 5 = 0 \]
\[ D = (-7)^2 - 4\cdot1\cdot(-5) = 49 + 20 = 69 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{69}}{2} \]
\[ x = \frac{7 + \sqrt{69}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{69}}{2} \]

The parabola y = x² - 7x - 5

xy
0.5-8.25
1.5-13.25
2.5-16.25
3.5-17.25
4.5-16.25
5.5-13.25
6.5-8.25

Questions

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

The solutions of x² - 7x - 5 = 0 are x = (7 + √69)/2 or x = (7 - √69)/2, that is x ≈ 7.6533119 and x ≈ -0.65331193.

Does x² - 7x - 5 = 0 factor?

Not over the integers, because the discriminant 69 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 7x - 5?

(7/2, -69/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7, product = c/a = -5.

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    Solve x² - 7x - 5 = 0: x = (7 + √69)/2 or x = (7 - √69)/2 (steps) | WordGPT