How do you solve the following quadratic equations methods:factoring, completing the square, quadratic formula

Question

1. 3x2 + 11x – 20 = 0

2. x2 + 3x – 4 = 0

3. 3x2 – x – 1 = 0

Answer 1

First try factorising. If you cannot get that to work, use one of the other two methods. The formula is faster when you have had practice in using it, but completing the square gives you a better understanding of what the formula achieves.

You should be able to factorise the first two, as b^2 - 4ac is a perfect square. For the third one, it isn't.

Answer 2

Hi,

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For the sake of easy reference , let us denote all of the quadratic equations as ax^2 + bx + c = 0

Then the discriminant D = b^2 - 4ac
For the equation to be factorized over integers, the D should be a perfect square. (as already posted by 'Rackbrane')
Otherwise use the quadratic formula.
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3) 3x2 – x – 1 = 0
here D = (-1)^2 - 4(3)(-1) = 13 (not a square)
use quadratic formula
x = [-b +/- sqrt(D)]/2a

1) 3x^2 + 11x – 20 = 0
Here a = 3, b = 11 and c = - 20
The discriminant b^2 - 4ac = 121 + 240 = 361 = (19)^2
It can be factorized as follows:

As ac = - 60, which is a negative number
which means we have to find the factors for - 60 whose difference is 'b' which is + 11.
The following factors of - 60 are indicated in brackets with their difference outside
(60, -1):59, (30, -2):28, (20, -3):17, (15, -4):11
So 15 and - 4 are the factors for - 60
Complete the factorization as posted by 'grompfet'

2)x2 + 3x – 4 = 0
here D = 9 + 16 = 25 = 5^2, Hence can be factorized
ac = - 4 and b = + 3
factors are
(4, - 1):3
x^2 + 4x - x - 4 = 0
follow steps as answered by 'grompfet'
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Note:
1)if ac = +ve Number, then the sum of the factors of ac = b and the sign of each factor is same as sign of b.
2)For a quadratic to be solved using completion of squares, then 'a' and 'c' have to have same signs and each should be a perfect square. In all the 3 problems a & c are not perfect squares.
3) whatever the type of quadratic, quadratic formula is a gun-shot method to solve.
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Good luck

Answer 3

Detailed discussion on x^2 + 2x - 5 = 0 ... whether it is a complete square or not As apparent, x^2+2x-5 is not a complete square. The complete square is in the format of x^2 + 2xy + y^2 ..... or x^2 - 2xy + y^2 Given it the shape of quadratic equation, we expect to have a^2x^2 + bx + c^2 = 0 ... such that {b = 2*a*c} In the given case: Let x^2 + 2x - 5 = a^2x^2 + bx + c^2 We get .......... a=1 and b=2=2*1*1, c=-5 (expected value should be 1^2=+1) Let's work out the difference: (x^2 + 2x + 1) - (x^2 + 2x - 5) = 6 Well, To get a complete square in place of it we should add 6 in it x^2 + 2x - 5 + 6 = x^2 + 2x +1 = (x + 1)^2 Our answer is: What do we add to have a complete square? .......... 6 What do we get then? ........... (x + 1)^2 ...................

Answer 4

1. (3x-4)(x+5)=0; x= 4/3, -5

That one is by factoring simply by trying different combinations of factors.

2. (x+4)(x-1)=0; x=-4,1

Again, trial and error with factoring will get this one.

3. 3x^2 -x -1. Factoring will not work for this one (no options FOIL out to the original polynomial). For quadratic formula, in this case, a=3, b=-1, and c=-1 as well.

x=[-b+/- (b^2-4ac)^(1/2)]/(2a), so [1 +/- (1+12)^(1/2)]/6. This simplifies to [1+/- (13)^(1/2)]/6 or 1 plus or minus the square root of 13 all divided by 6. Those are rather nasty answers, though.

Answer 5

1. (3x-4)(x+5) = 0

x=-5 or x=4/3

2. (x+4)(x-1) = 0

x=-4 or x=1

3. doesn't factor, and it's difficult to just complete the square since you'd have to divide everything by three, which makes fractions, so quadratic equation is probably best:

x = 1 +/- sqrt(1-4*3*(-1)) / 2*3

x = 1 +/- sqrt(1+12) / 6

x = 1 +/- sqrt(13) / 6

x = (1/6) +/- sqrt(13) / 6

Answer 6

1.) 3x^2 + 15x - 11x - 20 = 0
3x(x + 5) - 11 (x + 5) = 0
(3x - 11) (x + 5) = 0
x = 11/3 , -5

2.) x^2 + 4x - x - 4 = 0
x(x + 4 ) -1 ( x + 4) = 0
x = 1, -4

3.) 3x^2 - x - 1 = 0
use quadratic formula,

x = [1 +/- (13)^1/2]/ 6

Answer 7

Question 1
(3x - 4) (x + 5) = 0
x = 4 / 3 , x = - 5

Question 2
(x + 4) (x - 1) = 0
x = - 4 , x = 1

Question 3
x = [ 1 ±√(1 + 12) ] / 6
x = [1 ± √13] / 6
x = 0.768 , x = - 0.434

Answer 8

quadratic formula is

-b+ or - (square root) b^2 -4 a *c / 2 *a