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What is the root of the equation m squared -6m -7 equals 0 found by completing the square?
To complete the square for a quadratic ax² + bx + c = 0
1) Divide through by the coefficient of the x² term to get x² + b/a x + c/a = 0; let B = b/a; let C = c/a; the equation is now x² + Bx + c = 0
2) Complete the square by taking half of B and adding it to x and squaring: (x + B/2)²
3) Expanding this gives: (x + B/2)² = x² + Bx + (B/2)² → x² + Bx = (x + B/2)² - (B/2)²
4) Substitute this in the original equation to give: x² + BX + C = (x + B/2)² - (B/2)² + C
5) This can now be solved as (x + B/2)² - (B/2)² + C = 0 → (x + B/2)² = (B/2)² - C
For m² - 6m - 7 = 0, completing the square gives:
(m - 6/2)² - (6/2)² - 7 = 0
→ (m - 3)² - 9 - 7 = 0
→ (m - 3)² -16 = 0
→ (m - 3)² = 16
→ m - 3 = ±4
→ m = 3 ± 4
→ m = 7 or -1.
The coefficient of m is -6 so you need to use -6/2 = -3 for completing the squares.
m^2 - 6m - 70
=> m^2 - 6m = 70
=> (m - 3)^2 = 70 + (-3)^2 = 70 + 9
=> (m - 3)^2 = 79
=> m - 3 = +/- sqrt(79)
=> m = 3 +/- sqrt(79)
I think you want: m² - 6m - 70 = 0. Complete the square, so if you have something like
(x - A)², then you will get x² - 2Ax + A². So let's look at it. The coefficient of x is -6, so if -6x equates to -2Ax, then A must be 3.
So we have (m - 3)² = m² - 6m + 9, except you have m² - 6m - 70, so to get it like that we can do this: m² - 6m + 9 - 79 will equal your expression (9 - 79 = -70).
m² - 6m + 9 - 79 = 0. Add 79 to both sides. m² - 6m + 9 = 79
Now remember that m² - 6m + 9 = (m - 3)², so we have (m - 3)² = 79.
Square root of both sides: m - 3 = sqrt(79) and also: m - 3 = -sqrt(79).
Add 3 to both sides: m = 3 + sqrt(79) and m = 3 - sqrt(79)
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