solve the equation by completing the square

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jemma_1610
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hi it'd be a great help if anyone could help me work out the solutions and answers to these problems

1)
a) the quadratic x² + 4x - 21 can be written in the form of (x + a)² + b

find the value of a and b

b) hence or otherwise, solve the equation x² + 4x - 2 = 0


2) solve the equation 2x² + 8x - 1 = 0
by completing the square.
give your answers correct to 2 decimal places

3) the solutions of the equations x² + 6x + 4 = 0 can be written as ± √b where a and b where prime numbers
solve x² + 6x + 4 = 0 to find the values of a and b

4)a) the quadratic 3x² - 12x - 11 = 0 can be written in the form of 3(x + a)² + b
find the values of a and b
b) given that the solutions of the equations 3x² - 12x - 11 = 0 can be written as c ± √d , where c and d are rational numbers
find the value of a and d
note : please write your answer for d as a fraction in its simplest form
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mqb2766
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I think you have 2 similar posts and something like
https://www.mathsisfun.com/algebra/c...ng-square.html
should give you a review as well as the videos in the other thread.

Why is completing the square important? It put the quadratic in a standard form which
* makes it easy to sketch the graphs
* its 1/2 way to the usual quadratic equation x = -b+/-sqrt(b^2 ....
* it makes the quadratic solving process visual
so it is worth putting the effort into understanding it

1)
a) the quadratic x² + 4x - 21 can be written in the form of (x + a)² + b

Before solving this question, the minimum of the graph occurs at
x = -a
and the minimum value of "y" is b. If b is negative, the equation can be solved to give two roots, if b=0, the quadratic skims the x-axis and there only one real solution (repeated) and if b is positive, there are no real solutions.

The 4x term gives "a". Simply half the coefficient so a=2. So this gives a quadratic of
(x+2)^2 + b = x^2 + 4x + 4 + b = x^2 + 4x - 21
so b must be -25. -21 has been changed by -a^2 to give b.

So I know:
* The minimum of the graph occurs at x = -2, and the value of y at this point is -25.
* The roots are symmetric around x = -2. Its not too hard to see they're 3 and -7 as 5^2 = 25.

Have a look at the mathsisfun (and others) pages to make sure you understand this and try the other questions and post your working if/when you get stuck.
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