solve thee following quadratic equations by completing the suare

Question

x^2+8x+60=0

4x^2+6x-65=0

3x^2-6x+13=0

x^2+6x=59

5x^2-25=10x

Answer 1

I see that the question says to solve 3 of the 5 quadratic equations by completing the square. My guess is that 2 of them do not have real number solutions and you're suppose to solve the other 3 that do.

Put your equation in the form x² + bx = c

Noting that (x + b/2)² = x² + bx + b²/4, you can write

(x + b/2)² = c + b²/4

This means that you can take the square root of both sides (don't forget the ±) to arrive at

x + b/2 = ±√(c + b²/4)

and finally (after subtracting b/2 from both sides) you arrive at your answer

x = –b/2 ±√(c + b²/4)

So x² + 8x + 60 = 0 gets rewritten as x² + 8x = –60

which means that (x + 4)² = –60 + 16 = –44

For this one you'll be taking the square route of a negative number so your solutions will be complex (not real) numbers. (If my guess above is correct the this would be one of the two that you don't solve.)

In the second case

4x² + 6x – 65 = 0 gets rewritten as 4x² + 6x = 65

then again (after dividing both sides by 4) as x² + (3/2)x = 65/4

which means that (x + 3/4)² = 65/4 + 9/16 = (260 +9)/16 = 269/16

and you finish by

taking the square root of both sides (don't forget the ±), then

subtracting 3/4 from both sides.

Of the remaining three quadratic equations, one will have complex roots and the other two will have real roots.

I hope this gives you some help understanding how to tackle the assigned questions.

Answer 2

Hi Destiny B.

First you want to review the requirements and rules to solve by Completing the Square you can find these in your text or online: they include the following:

In f(x) = ax² + bx + c, a must be 1

Next you want to get the x terms on one side of the equation and c, the constant on the other side

To create a trinomial Square add (b/2)² to both sides of the equation

Finally solve for x, keeping in mind everything you’ve learned about numbers as needed

I will do a couple below:

x² + 8x + 60 = 0

a = 1

x² + 8x= -60

b = 8 so b/2 = 4

x² + 8x + (4)² = -60 + (4)²

(x + 4)² = -44

(x + 4)² = (-1)(44)

x + 4 = ± (√-1)(√44)

√(-1) = i

√44 = ±(√4)(√11)

x + 4 = ±( i*2*√11) = ±2i√11

x = -4±2i√11

5x² -25 = 10x

a = 5, so we need to divide both sides of the equation by 5

x² - 5 = 2x

Collect the x terms on side and the constant on the other

x² - 2x = 5

b = -2, (b/2) = -1

x² - 2x + (-1)² = 5 + (-1)²

x² - 2x + 1 = 6

(x - 1)² = 6

x - 1 = ±√6

Solve for x

x = 1±√6

Try the rest on your own and remember to review the requirements and rules for Completing the Square.

If you have any questions, write back, or seek out a tutor.

I hope this helps