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)