Respuesta :
y = 2x^2 + 5x - 12 is in the form y = ax^2+bx+c
We can line up the terms to see this
y = 2x^2 + 5x - 12
y = ax^2 + bx + c
We see that
a = 2
b = 5
c = -12
Plug this into the discriminant formula
d = b^2 - 4ac
d = 5^2 - 4*2*(-12)
d = 25 - 8*(-12)
d = 25 + 96
d = 121
The discriminant is a positive whole number, and it is a perfect square. Note how 11*11 = 11^2 = 121
Taking the square root of 121 will lead to a whole number (11), which tells us that 2x^2 + 5x - 12 can be factored by grouping. In general, if the discriminant is a perfect square, then the quadratic can be factored.
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Let's factor by grouping
Multiply the first coefficient 2 by the last term -12 to get -24
Now ask yourself: "What two numbers multiply to -24 and add to 5?"
The question to that answer is "the numbers -3 and 8"
Based on that answer, we can break the 5x into -3x+8x. Note how -3x+8x adds back up to 5x
2x^2 + 5x - 12
2x^2 - 3x + 8x - 12
(2x^2 - 3x) + (8x - 12)
x(2x - 3) + (8x - 12)
x(2x - 3) + 4(2x - 3)
(x + 4)(2x - 3)
So 2x^2 + 5x - 12 completely factors to (x + 4)(2x - 3)
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If you don't like the factor by grouping method, then you can use the quadratic formula to find that the two solutions are
x = -4 or x = 3/2
Then work the zero product property backwards to set up the factors
x = -4 or x = 3/2
x = -4 or 2x = 3
x+4 = 0 or 2x-3 = 0
(x+4)(2x-3) = 0
and we get the same result as before.
We can line up the terms to see this
y = 2x^2 + 5x - 12
y = ax^2 + bx + c
We see that
a = 2
b = 5
c = -12
Plug this into the discriminant formula
d = b^2 - 4ac
d = 5^2 - 4*2*(-12)
d = 25 - 8*(-12)
d = 25 + 96
d = 121
The discriminant is a positive whole number, and it is a perfect square. Note how 11*11 = 11^2 = 121
Taking the square root of 121 will lead to a whole number (11), which tells us that 2x^2 + 5x - 12 can be factored by grouping. In general, if the discriminant is a perfect square, then the quadratic can be factored.
---------------------------------------------------
Let's factor by grouping
Multiply the first coefficient 2 by the last term -12 to get -24
Now ask yourself: "What two numbers multiply to -24 and add to 5?"
The question to that answer is "the numbers -3 and 8"
Based on that answer, we can break the 5x into -3x+8x. Note how -3x+8x adds back up to 5x
2x^2 + 5x - 12
2x^2 - 3x + 8x - 12
(2x^2 - 3x) + (8x - 12)
x(2x - 3) + (8x - 12)
x(2x - 3) + 4(2x - 3)
(x + 4)(2x - 3)
So 2x^2 + 5x - 12 completely factors to (x + 4)(2x - 3)
---------------------------------------------------
If you don't like the factor by grouping method, then you can use the quadratic formula to find that the two solutions are
x = -4 or x = 3/2
Then work the zero product property backwards to set up the factors
x = -4 or x = 3/2
x = -4 or 2x = 3
x+4 = 0 or 2x-3 = 0
(x+4)(2x-3) = 0
and we get the same result as before.
