Unit 4: Radicals and the Quadratic Formula

multiplying conjugates

(a + b)(a - b) = a^2 - b^2 (conjugate: formed by changing the sign between two terms in a binomial --> one will be + the other will be - )

The SUM of the roots of a quadratic is

-b/a

How to solve radical equations

1.) Isolate the radical (to one side of the equation)... 2.) Square both sides (to get rid of the radical symbol)(*Double distributed if there's a binomial*)... 3.) Solve the resulting equation... 4.) Check for extraneous solutions (numbers that do not satisfy the original equation, they are not solutions to the original equation... can have no solution)

Every quadratic equation has

2 roots (zeroes): x1 = -b *+* √(b² - 4ac)/2a AND x2 = -b *-* √(b² - 4ac)/2a

perfect square

A rational number whose square root is a whole number

Zero-Exponent Property

Any non-zero number raised to the zero power equals 1 (a^0 = 1)

The quadratic formula can be used to solve

Any quadratic equation

Don't forget to

DOUBLE DISTRIBUTE algebraic expressions (ex: √2x-3 = x-3 --> square it but make sure to use FOIL/double distribute the x-3 --> 2x-3 = (x-3)(x-3))

Power of a Product Property

Find the power of *each* factor in the parenthesis and multiply [(ab)^m = a^m • b^m]

Negative Exponent Property

For any non-zero base raised to a negative exponent, place the power in the denominator to re-write the power with a positive exponent (x^-n = 1/x^n)

Power of a Quotient

For any number "a" and "b", where b is not zero, if the quotient of "a" and "b" is raised to the power, raise both the numerator and the denominator to the given power [(a/b)^m = a^m/b^m] (quotient = an answer to a division problem)

Quotient Property

If dividing two numbers with the *same* base, keep the base and SUBTRACT the exponents (x^a/x^b = x^a - b)

Product Property

If multiplying two numbers with the *same* base, keep the base and ADD the exponents (x^a • x^b = x^a + b)

Dividing Radicals

If possible, divide coefficients, divide radicands, and then simplify (subtract exponents) (ex: √20x^3 / √5x --> √20x^3/5x --> √4x^2 --> 2x.... ex:

Power to Power Property

If raising a power to a power, MULTIPLY the exponents [ (x^n)^m = x^n • m]

The domain of a square root function is

RESTRICTED because the square root of a negative number does not produce a REAL number.

Simplifying radicals

Re-write each variable as the largest perfect square factor (even exponent) and simplify (ex: √27x^11y^5 --> √9x^10y^4 & √3xy --> 3x^5y^2√3xy)

Radicals can always be re-written using fractional exponents:

X^(power/root) = r√x^p (if there's nothing written before the radical, then its a 2 or square root)

reciprocal

a flipped fraction

square root function

a function that contains a square root of a variable. f(x)=√x

standard form of a quadratic equation

ax² + bx + c = 0

The PRODUCT of the roots of a quadratic is

c/a (think "c" in product = c/a)

To use the square root method on a quadratic equation WITH a linear term

complete the square, then solve (ex: x^2 - 10x + 23 = 0 --> subtract 23 --> x^2 - 10x + ___ = -23 + ___ --> (b/2)^2 and add it to both sides --> x^2 - 10x + 25 = -23 + 25 --> (x-5)^2 = 2 --> take square root --> √(x-5)^2 = √2 --> x-5 = +/- √2 --> add 5 --> x = 5 +/- √2)

When we square both sides of an equation, there is a possibility that we can introduce

extraneous solutions ("extra" solutions) so check your solutions by plugging them back into the equation.

Multiplying Radicals

multiply coefficients, multiply radicands, then simplify (add exponents) (ex: 3y√2x • 4√6x^3 --> 12y√12x^4 --> 12y, √4x^4 & √3 --> 12y • 2x^2 √3 --> 24x^2y√3 )

To rationalize a fraction with a BINOMIAL denominator

multiply the numerator and denominator by the CONJUGATE of the denominator (ex: 4/√6 + 5 • √6. - 5 / √6. - 5 --> 4√6 - 20 / √6^2 - 5^2 --> 4√6 - 20 /-19)

to rationalize a fraction with a MONOMIAL denominator

multiply the numerator and denominator by the radical expression in the denominator (ex: 4/√2 • √2/√2 --> 4√2/2 --> 2√2) (simplify numerator if you can)

We can use the SQUARE ROOT METHOD when there is

no LINEAR TERM in a quadratic function (aka no middle term, no term with an x, the b is zero). Do this by getting x^2 by itself and taking the square root. (remember to put the +/- sign before your answer) (if there is a negative under the radical, there is NO REAL SOLUTION, the number is imaginary) (ex: 5x^2 - 7 = 12 --> 5x^2 = 19 --> divide by 5 --> x^2 = 19/5 --> √x^2 = √19/5 --> x = +/- √19/√5 • √5/√5 --> x = +/- √95/5)

A radical can be simplified if the

radicand (the number underneath the radical) has a factor that is a perfect square (ex: √72 --> √36 & √2 --> 6√2 ...... ex: √x^6y^9 --> √x^6y^8 & √y --> x^3y^4√y) (only *even exponents* can be simplified --> divide them in half)

To find the domain of a square root function

set the inside of the radical to greater than or equal to zero (ex: y = √x+2 --> x + 2 ≥ 0)

Cube roots, fourth roots, fifth roots, etc can be

simplified just like square roots can be (instead, just find the perfect cube, perfect fourth, perfect fifth etc that is a factor of the radicand)

A radical multiplied by the same radical equals

the inside of the radical (√x • √x = x .... √6 • √6 --> √36 = 6)

Radicals can only be added/subtracted if

the radicands are the SAME (ex: x√2 + 3x√2 --> 4x√2 .... ex: √8 - √18 --> √4 & √2 , √9 & √2 --> 2√2 - 3√2 --> -√2)

A number is in simplest radical form when

there is NO radical in the DENOMINATOR

Quadratic Formula

x = -b ± √(b² - 4ac)/2a (keep +/- symbol)(simplify everything that you can)