### March 2018 archive

While solving quadratic equations, it’s useful to take a look at the discriminant.

The discriminant is area under the square root in the quadratic formula: if you have a quadratic equation (equation equal to zero with 3 distinct parts), you can use the quadratic formula to solve. Depending on the answer, we can figure out whether the equation will have 1,2 or 0 solutions.

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Let’s find the discriminant of the equation: $x^2-3x+6=0$

In a quadratic equation, the parent would be $ax^2+bx+c$

following the parent, in our equation, a=1, b=-3, and c=6

using this information we can plug in the numbers to our equation to find the discriminant.

If $b^2-4ac$ is our equation, we just put our numbers we found in the spots of the letters, and simplify. $b^2-4ac$ $-3^2-4(1)(6)$ $9-4(6)$ $9-24$ $-15$

using this we know that since the discriminant is -15, the original quadratic equation does not have any solutions, and using the quadratic formula would not work.

This week we learned how to solve a quadratic equation using factoring.

In order to solve the equation the zero product law must be used.

a*b=0, so either a=0 or b=0.

both sides if the equals sign must be equal.

for example: 8(x+2)(x-7)=0

x+2=0

so x=-2

also, x-7=0

so x=7

a quadratic equation has two possible answers. If you plug either of them into the equation, it should work.

8(x+2)(x-7)=0

x=-2

8(-2+2)(-2-7)=0

8(0)(-9)=0

8(0)=0

0=0

to solve any quadratic equation, the product must be 0. The equation cannot work without it.

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When factoring polynomials, we can use a system:

Common?

Difference of squares (2 terms)

Pattern ( $x^2$ x #) (3 terms)

Easy $1x^2$

Ugly $ax^2$

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using these steps we can factor each type of polynomial to its simplest form.

for example:

3y(y+2) – 9(y+2)

because this polynomial has a common factor, y+2, we can substitute it for an unknown variable such as “a”

3ya – 9a

doesn’t that look better?

now we can use the to find the difference of squares because there’s 2 terms. If there are more than two terms you can skip the and go right to P.

3a is common on both terms so we simplify polynomial to

3a(y-3)

now we substitute our y+2 back for “a”.

3(y+2)(y-3)

and you’re done! this polynomial cannot be simplified farther.

For a polynomial with 3 terms, you will not use the D, but skip right to P depending on whether you have $1x^2$ or $ax^2$ will decide if you use the E or the U.