Problem sET
Verbal
1. How do we recognize when an equation is quadratic?
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It is a second-degree equation (the highest variable exponent is 2).
2. When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?
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We want to take advantage of the zero property of multiplication in the fact that if a⋅b=0 then it must follow that each factor separately offers a solution to the product being zero:a=0 or b=0.
3. In the quadratic formula, what is the name of the expression under the radical sign and how does it determine the number of and nature of our solutions?
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It is called a discriminant. Its sign lets us determine the number and the type of solutions the equation has.
Algebraic
For the following exercises, solve the quadratic equation by factoring.
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x=6,x=3
5. 6x2+17x+5=0
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x=−52,x=−13
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x=5,x=−5
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x=−32,x=32
8. [latex]5{x}^{2}=5x+30[/latex]
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x=−2,x=3
9. [latex]7{x}^{2}+3x=0[/latex]
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x=0,x=−37
For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.
11. [latex]9{x}^{2}-30x+25=0[/latex]
12. [latex]6{x}^{2}-x-2=0[/latex]
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Two real; rational
For the following exercises, solve the quadratic equation by using the quadratic formula.
13. [latex]{x}^{2}+x=4[/latex]
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x=−1±√172
14. [latex]3{x}^{2}-5x+1=0[/latex]
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x=5±√136
15. x2=−25
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{5i,−5i}
16. x2+36=0
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{2i√2,−2i√2}
17. x2+2x+5=0
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{3i√3,−3i√3}
18. x2+8x+25=0
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{2+i,2−i}
19. x2+6x+25=0
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{2+3i,2−3i}
20. x2−6x+10=0
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{5+i,5−i}
21. x(x−2)=10
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{2+2√6,2−2√6}
22. 5x2−8x+5=0
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{−12+32i,−12−32i}
23. 2x2−6x+5=0
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{−35+15i,−35−15i}
24. x2−2x+4=0
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{−12+12i√7,−12−12i√7}