Quadratic Equations

Solving a Cubic Equation

1. Factor as Sum of Cubes 2. Zero-Factor Property 3. Quadratic Formula 4. Simplify 5. Factor Out 6. Lowest Terms

Quadratic equations in one variable

*ax²+bx+c=0* where a and b are real numbers and *a≠0*

Zero-Factor Property

*if a and b are complex numbers with ab=0, then a=0 or b=0 or both.* exp. 6x²+7x=3 6x²+7x-3=0............................................Standard form (3x-1)(2x+3)=0..................................................Factor 3x-1=0 or 2x+3=0..........Zero-factor property 3x=1 or 2x=-3 x=1/3 or x=-2/3............................................Solution set

Square Root Property

*if x²=k, then x=√k, or x=-√k* solution set *{√k,-√k}* or *{±√k}* both solutions are real if *k>0*, and both are pure imaginary if *k<0*. if *k<0* then *{±i√|k|}*.(if *k =0 *then there is only one distinct solution, 0, sometimes called a *double solution*.)

Quadratic Formula

*x=[-b±√(b²-4ac)] / (2a)* *ax²+bx+c=0* where *a≠0*

Solving a Quadratic Equation by Completing the Square

if (ax-b)²=c then ax-b= ±√c then ax=b±√c if c = d+e² or c=d²+e² then ax=b±e√d or ax=b±ed then x=(b±√c)/a or x=(b±e√d)/a or x=(b±ed)/a the solution set for x=(b±ed)/a is {(b+ed),(b-ed)} x=(b±√c)/a={(b-√c),(b+√c)} x=(b±e√d)/a={(b-e√d),(b+e√d)}

Quadratic Equation

is a second-degree equation—that is, an equation with a squared variable term and *no terms of greater degree!* Its general format is ax²+bx+c=0, where x represents a variable and a, b, and c represent coefficients and constants, where a≠0 (If a=0, the equation becomes a linear equation.). The constants a, b, and c are called the quadratic coefficient, the linear coefficient and the constant term or free term respectively. (The term quadratic comes from quadratus, which is Latin for "square.")

Solving for a Variable in a Formula

Example. A=(πd²)/(4) 4A=πd²......................................multiplied by denominator (4A)/(π)=d².........................................................divided by π d=±√[(4A)/(π)]....................................square root property d=[(±√(4A))/(√π)]•[(√π)/(√π)]....................... rationalize denominator d=[±√(4Aπ)]/(π)............................................... multiply numerators/denominators d=[±2√(Aπ)]/(π)..................................simplify

Solving for a Variable in a Formula

Example. rt²-st=k (r≠0), for t rt²-st-k=0........................................................standard form t=[-b±√(b²-4ac)]/(2a)..............................Quadratic formula t=[-(-s)±√(-s²-4r(-k))]/(2r)..................................a=r,b=-s,c=-k t=[s±√(s²+4rk)]/(2r)..................................................simplify

Cubic Equation

Exp. *x³+8=0*

Note for solving for variables.

In examples solving for a variable, we took the *positive and negative* square roots. However, if the variable represents a distance or length in an application, we would consider *only the positive* square root.

The Discriminant

The quantity under the radical in the quadratic formula, [-b±√(*b²-4ac*)]/(2a) again, *b²-4ac*, it is called the *discriminant.*

Solving for a Specified Variable

To solve a quadratic equation in a specified variable in a formula or in a literal equation, we usually apply the *Square Root Property* or the *Quadratic Formula.*

The Discriminant. Note

When the numbers a, b, and c are *integers* (but not necessarily otherwise), the value of the discriminant can be used to determine whether the solutions of a quadratic equation are rational, irrational, or nonreal complex number.