. High school algebra . 2mx=n. Complete the square by adding m to each side, x* + 2mx-{-m* = n + m*.Take the square root, x+Tn= ± Vn+m*, x= âm+ Vn-f-w*.The two roots are âm+Vn+vi*, âmâVn-^m*. Ex. 2.âSolve z^-\-pz-\-q^O. Transpose the absolute term, x*+px= âq. p* p* p* p*â4o Add ~- to each side, x* + px + ^ â â 9 + ir â â^â 4 4 4 4 Take the square root, x + 2=± 2 -p Vp*-4q X = -t:â ± ;; THE GENERAL QUADRATIC EQUATION Ex. 3.âSolve aa;2+6a;+c=0. Divide by a to make the first term a square. 26& Transpose, r* + -r + - = a X* -\âX â â Add -râ. to each, x* Hâx -\- -râz = fe*â4ac 4a» Take the
. High school algebra . 2mx=n. Complete the square by adding m to each side, x* + 2mx-{-m* = n + m*.Take the square root, x+Tn= ± Vn+m*, x= âm+ Vn-f-w*.The two roots are âm+Vn+vi*, âmâVn-^m*. Ex. 2.âSolve z^-\-pz-\-q^O. Transpose the absolute term, x*+px= âq. p* p* p* p*â4o Add ~- to each side, x* + px + ^ â â 9 + ir â â^â 4 4 4 4 Take the square root, x + 2=± 2 -p Vp*-4q X = -t:â ± ;; THE GENERAL QUADRATIC EQUATION Ex. 3.âSolve aa;2+6a;+c=0. Divide by a to make the first term a square. 26& Transpose, r* + -r + - = a X* -\âX â â Add -râ. to each, x* Hâx -\- -râz = fe*â4ac 4a» Take the square root, 4a« ^ + 2^=± 4a* a 4a^ v62_4ac 2a h . V6*â4ac .. aJ = â K- + ;j 2a - 2a _ -6±\/&2â4ac~ 2a The roots of the general quadratic equation are â &± V&^â4ctc2a 188. The roots of the general quadratic might also be bund by factoring as in art. 171. ax*-[-hx-\-c = 0, \ a a] (f , by^c 6M. Since the product is zero, one of thea is not zero, as the equation would not then ^ + 2^ + must be zero. Butquadratic. 270 ALGEBRA eixbrcise: 127 Solve by either of the preceding methods :1.* a:»-2aa:=3a2. 2. a;2+46z-56*=0. 3. z*â6mx-H3m2=0. 4. x^+4:pxâp^=(i. 6. z*-2az+6=0. 6. x^^2bx-c=0. 7. o«»+2ax=6. 8. ax2-i-26x+c= az*â6zâc=0. 10. px^âqx+r=(). 189. Solving by Formula. The roots of any particularquadratic equation may be found b} substituting the valuesof a, h and c in the roots of the general quadratic. Ex. 1.âSolve 6a;2-7a;+2=0. Herea = 6, 6=-7, c = 2. â , . , , . â6+V6>-4ac Substitute these values in x= ^^ » 2a + 7±V49-48â¢â ⢠^= 12 7±1 8 6 2 1 = 12 =r2°r2 = 3°2- Verify by substitution. Ex. 2.âSolve 5a:2+6x-l=0. Here a=5, 6 = 6, c= â1. _ -6±\36-(-20) _ -6+^56â¢â¢ ^~ 10 ~ 10 -6±2Vl4 -3±VIi~ 10 ~ 5 â In this case the roots are irrational, but, if necessary, we maysubstitute for Vli its approximate value 3-742, when the roots become -3Â
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