BackClose

Select a Subject

Math

Question 3 of 18

beginning of content:

M:N:SPR:3

Tags:
SAT

Exam

Learn more about this exam.

PSAT/NMSQT

Exam

Learn more about this exam.

PSAT 10

Exam

Learn more about this exam.

Calculator: Not Permitted
Passport to Advanced Math

Subscore

Learn more about Passport to Advanced Math.

If a squared plus 14 a equals 51 and a greater than 0 comma what is the value of a plus 7 ?

Select an Answer

Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No
Correct Answer: 
No

View Correct Answer

10

There is more than one way to solve this problem. A student can apply standard techniques by rewriting the equation a squared plus 14 a equals 51 as a squared plus 14 a minus 51 equals 0 and then factoring. Since the coefficient of a is 14 and the constant term is minus 51 comma factoring a squared plus 14 a minus 51 equals 0 requires writing 51 as the product of two numbers that differ by 14. This is 51 equals open parentheses 3 close parentheses open parentheses 17 close parentheses comma which gives the factorization a squared plus 14 a minus 51 equals open parentheses a plus 17 close parentheses open parentheses a minus 3 close parentheses equals 0. The possible values of a are a equals minus 17 and a equals 3. Since it is given that a greater than 0 comma it must be true that a equals 3. Thus, the value of a plus 7 is 3 plus 7 equals 10.

A student could also use the quadratic formula to find the possible values of a :

a equals fraction numerator minus 14 plus-or-minus square root of 14 squared minus 4 open parentheses 1 close parentheses open parentheses minus 51 close parentheses end root over denominator 2 open parentheses 1 close parentheses end fraction
space space equals fraction numerator minus 14 plus-or-minus square root of 196 minus open parentheses minus 204 close parentheses end root over denominator 2 end fraction
space space equals fraction numerator minus 14 plus-or-minus square root of 400 over denominator 2 end fraction
space space equals fraction numerator minus 14 plus-or-minus 20 over denominator 2 end fraction

The possible values of a are a equals fraction numerator minus 14 minus 20 over denominator 2 end fraction equals minus 17 and a equals fraction numerator minus 14 plus 20 over denominator 2 end fraction equals 3. Again, since it is given that a greater than 0 comma it must be true that a equals 3. Thus, the value of a plus 7 is 3 plus 7 equals 10.

There is another way to solve this problem that will reward the student who recognizes that adding 49 to both sides of the equation yields a squared plus 14 a plus 49 equals 51 plus 49 comma or rather open parentheses a plus 7 close parentheses squared equals 100 comma which has a perfect square on each side. Since a greater than 0 comma the solution a plus 7 equals 10 is evident.

Question Difficulty: 
easy
Objective: 

Students must solve a quadratic equation.