X2= 12 + XTHIS EQUATION NEEDS TO BE PUT INTO STANDARD FORM BY SUBTR...

3. x

²

= 12 + xThis equation needs to be put into standard form by subtracting 12 and xfrom both sides of theequation. x

²

– x– 12 = 12 – 12 + x– xx

²

– x– 12 = 0

–

A L G E B R A

–

Since the sum of 3 and –4 is –1, and their product is –12, the equation factors to (x+ 3)(x– 4) = 0Set each factor equal to zero and solve: x+ 3 = 0 or x– 4 = 0x= –3 or x= 4The solution is {–3, 4}.By Quadratic FormulaSolving by using the quadratic formula will work for any quadratic equation, especially those that are not fac-torable.Solve for x:x

²

+ 4x= 1Put the equation in standard form. x

²

+ 4x– 1 = 0Since this equation is not factorable, use the quadratic formula by identifying the value ofa,b, andcand then substituting it into the formula. For this particular equation,a = 1,b= 4, and c= –1.x –b ; 2 b

²

– 4ac2ax –4 ; 2 4

²

41121–122112x –4 ; 2 1642x –4 ; 2 2022 5x –42 ^;x –2 ; 2 5–2 2 5, –2 – 2 5 The solution is .The following is an example of a word problem incorporating quadratic equations:A rectangular pool has a width of 25 feet and a length of 30 feet. A deck with a uniform width sur-rounds it. If the area of the deck and the pool together is 1,254 square feet, what is the width ofthe deck?

3 4 6

Begin by drawing a picture of the situation. The picture could be similar to the followingfigure.

x

30

25

Since you know the area of the entire figure, write an equation that uses this information. Since we aretrying to find the width of the deck, let x= the width of the deck. Therefore,x+ x+ 25 or 2x+ 25 is the widthof the entire figure. In the same way,x+ x+ 30 or 2x+ 30 is the length of the entire figure.The area of a rectangle is length ×width, so use A = l ×w.Substitute into the equation: 1,254 = (2x+ 30)(2x+ 25)Multiply using FOIL: 1,254 = 4x

²

+ 50x+ 60x+ 750Combine like terms: 1,254 = 4x

²

+ 110x+ 750Subtract 1,254 from both sides: 1,254 – 1,254 = 4x

²

+ 110x+ 750 – 1,2540 = 4x

²

+ 110x– 504Divide each term by 2: 0 = 2x

²

+55x– 252Factor the trinomial: 0 = (2x+ 63 )(x– 4)Set each factor equal to 0 and solve 2x+ 63 = 0 or x– 4 = 02x= –63 x= 4x= –31.5Since we are solving for a length, the solution of –31.5 must be rejected. The width of the deck is 4 feet.

R a t i o n a l E x p r e s s i o n s a n d E q u a t i o n s

Rational expressions and equations involve fractions. Since dividing by zero is undefined, it is important toknow when an expression is undefined.You may be asked to perform various operations on rational expressions. See the following examples.

Examples

x

²

b