Answer:
Part 1) [tex]a^{2} -9a+14=0[/tex] -----> solution set {7,2}
Part 2) [tex]a^{2} +9a+14=0[/tex] -----> solution set {-2,-7}
Part 3) [tex]a^{2} +3a-10=0[/tex] -----> solution set {2,-5}
Part 4) [tex]a^{2} +5a-14=0[/tex] ----> solution set {2,-7}
Part 5) [tex]a^{2} -5a-14=0[/tex] ----> solution set {-2,7}
Step-by-step explanation:
we know that
The formula to solve a quadratic equation of the form
[tex]ax^{2} +bx+c=0[/tex]
is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
Part 1)
in this problem we have
[tex]a^{2} -9a+14=0[/tex]
so
[tex]a=1\\b=-9\\c=14[/tex]
substitute in the formula
[tex]a=\frac{9(+/-)\sqrt{-9^{2}-4(1)(14)}} {2(1)}[/tex]
[tex]a=\frac{9(+/-)\sqrt{25}} {2}[/tex]
[tex]a=\frac{9(+/-)5} {2}[/tex]
[tex]a=\frac{9(+)5} {2}=7[/tex]
[tex]a=\frac{9(-)5} {2}=2[/tex]
The solution set is {7,2}
Part 2)
in this problem we have
[tex]a^{2} +9a+14=0[/tex]
so
[tex]a=1\\b=9\\c=14[/tex]
substitute in the formula
[tex]a=\frac{-9(+/-)\sqrt{9^{2}-4(1)(14)}} {2(1)}[/tex]
[tex]a=\frac{-9(+/-)\sqrt{25}} {2}[/tex]
[tex]a=\frac{-9(+/-)5} {2}[/tex]
[tex]a=\frac{-9(+)5} {2}=-2[/tex]
[tex]a=\frac{-9(-)5} {2}=-7[/tex]
The solution set is {-2,-7}
Part 3)
in this problem we have
[tex]a^{2} +3a-10=0[/tex]
so
[tex]a=1\\b=3\\c=-10[/tex]
substitute in the formula
[tex]a=\frac{-3(+/-)\sqrt{3^{2}-4(1)(-10)}} {2(1)}[/tex]
[tex]a=\frac{-3(+/-)\sqrt{49}} {2}[/tex]
[tex]a=\frac{-3(+/-)7} {2}[/tex]
[tex]a=\frac{-3(+)7} {2}=2[/tex]
[tex]a=\frac{-3(-)7} {2}=-5[/tex]
The solution set is {2,-5}
Part 4)
in this problem we have
[tex]a^{2} +5a-14=0[/tex]
so
[tex]a=1\\b=5\\c=-14[/tex]
substitute in the formula
[tex]a=\frac{-5(+/-)\sqrt{5^{2}-4(1)(-14)}} {2(1)}[/tex]
[tex]a=\frac{-5(+/-)\sqrt{81}} {2}[/tex]
[tex]a=\frac{-5(+/-)9} {2}[/tex]
[tex]a=\frac{-5(+)9} {2}=2[/tex]
[tex]a=\frac{-5(-)9} {2}=-7[/tex]
The solution set is {2,-7}
Part 5)
in this problem we have
[tex]a^{2} -5a-14=0[/tex]
so
[tex]a=1\\b=-5\\c=-14[/tex]
substitute in the formula
[tex]a=\frac{5(+/-)\sqrt{-5^{2}-4(1)(-14)}} {2(1)}[/tex]
[tex]a=\frac{5(+/-)\sqrt{81}} {2}[/tex]
[tex]a=\frac{5(+/-)\sqrt{81}} {2}[/tex]
[tex]a=\frac{5(+)9} {2}=7[/tex]
[tex]a=\frac{5(-)9} {2}=-2[/tex]
The solution set is {-2,7}
