Casey W. answered • 06/16/15
Tutor
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Mathematics (and Science) Instruction by a Mathematician!
In general for a monic quadratic equation of the form:
x^2+bx=c, we can follow Mark's technique of writing
adding (b/2)^2 to both sides will make the left hand side a perfect square, (x+b/2)^2
x^2 + bx + (b/2)^2 = c + (b/2)^2
(x+b/2)^2 = c +(b/2)^2
And so we can then apply a square root (don't forget to introduce the +/-, obtained by expanding an absolute value), and subtracting b/2 from both sides:
(x+b/2) = +/- \sqrt{c+(b/2)^2}
x = -b/2 +/- \sqrt{c+(b/2)^2} (1)
x^2+bx=c, we can follow Mark's technique of writing
adding (b/2)^2 to both sides will make the left hand side a perfect square, (x+b/2)^2
x^2 + bx + (b/2)^2 = c + (b/2)^2
(x+b/2)^2 = c +(b/2)^2
And so we can then apply a square root (don't forget to introduce the +/-, obtained by expanding an absolute value), and subtracting b/2 from both sides:
(x+b/2) = +/- \sqrt{c+(b/2)^2}
x = -b/2 +/- \sqrt{c+(b/2)^2} (1)
Separate the constant terms and variable terms on either side of the equation, and multiply or divide by the leading coefficient to make the x^2 term have a coefficient of 1 in front (a monic equation)
2n^2+10n=-6 means n^2+5n = -3...and not that we can use b=5 and c=-3 in the formula (1) above.
z^2-14z = -34, means we can use b=-14 and c=-34 for the second problem!
Hope that helps!
