7 3 Solving Equations Using Quadratic Techniques Objectives

7. 3 Solving Equations Using Quadratic Techniques Objectives: 1. Write expressions in quadratic form. 2. Use quadratic techniques to solve equations.

Quadratic Form An expression that is quadratic in form can be written as au²+bu+c for any numbers a, b, and c, a≠ 0, where u is some expression in x. The expression au²+bu+c is called the quadratic form of the original expression. Example: Write each expression in quadratic form, if possible. 2 x⁶+x³+9 2(x³)²+(x³)+9 cannot be written in quadratic form

Solve each equation. x⁴-29 x²+100=0 Write as a quadratic equation. Let u=x² u²-29 u+100=0 Factor (u-25)(u-4)=0 Substitute x² back in (x²-25)(x²-4)=0 (x-5)(x+5)(x-2)(x+2)=0 x=5, -5, 2, -2 Solve x³+216=0 Factor as a sum of cubes (x+6)(x²-6 x+36)=0 x+6=0, x²-6 x+36=0 x=-6, use quad. formula to solve other equation

Recall about graphs/solutions Recall that the degree of an equation indicates the number of solutions to an equation. Some solutions will be imaginary, some will be real. Imaginary solutions always come in pairs. On the last example, the degree was 3 and there were 3 solutions, 1 real and 2 imaginary. Knowing this will ensure that you don’t leave out any solutions.

More solving equations Solve. Let

Check the solutions Plug both 81 and 16 back into original equation. The only solution is 81.

Solve Solving Radical Equations Set=0 Write in quadratic form. Check:

Homework: page 363 12 -28 even