Hard SAT Math Problem with Solutions

Yesterday I gave you a Level 5 SAT math problem. Today I will provide solutions for this problem. If you have not yet attempted the problem go back and take a look at it first so you can try it on your own. Here is the link: Hard SAT Math Problem

Level 5 – Passport to Advanced Math

Here is the problem once again followed by several solutions:

If x² – 8x = 209 and x < 0, what is the value of |x + 3| ?

Algebraic solution: Let’s attempt to factor the equation. We begin by subtracting 209 from each side to get x² – 8x – 209 = 0. We now factor the left hand side to get (x – 19)(x + 11) = 0.

So x = 19 or x = -11. Since we are given x < 0, we use x = -11.

Finally we have |x +3| = |-11 + 3| = |-8| = 8.

Notes: (1) There are several ways to solve a quadratic equation. A few are by (i) factoring, (ii) completing the square, (iii) the quadratic formula, (iv) guessing and checking, (v) creating a table of values in your calculator, (vi) using the graphing features of your calculator.

(2) Whenever you solve a quadratic equation by factoring or by using the quadratic formula, you need to bring everything over to one side of the equation first, leaving 0 on the other side.

(3) It can seem very difficult at first to find the factors of 209, but the following trick can help you find the factors a little easier. Since we have 15² = 225 > 209, it follows that if 209 can be factored, then it has a factor less than 15. Furthermore, since every positive integer can be factored as a product of primes, it follows that we need only check prime numbers less than 15. So we can simply check 209 for divisibility by 2, 3, 5, 7, 11, and 13.

We can use standard divisibility tricks to eliminate 2, 3, and 5 right away (if you don’t know these tricks you can learn about them here: Divisibility Tricks). You may want to try 11 next since it is pretty easy to divide by 11. In this case we have 209/11 = 19. So 209 = 11 ∙ 19.

If we happen to be allowed to use a calculator for this problem, then we could divide 209 by these numbers very quickly.

(4) Several more notes giving more insight into this solution can be found in New SAT Math Problems arranged by Topic and Difficulty Level.

Solution by guessing and checking: If we are allowed to use our calculator, there are several other ways to solve the given equation for . One way is to simply guess and check negative values for x. For example if we substitute -3 in for x on the left hand side of the equation we get (-3)^2 – 8(-3) = 33, which is too small. So let’s try x = -8 next: (-8)^2 – 8(-8) = 128 . This is still too small, but we’re heading in the right direction. We try x = -11 to get (-11)^2 – 8(-11) = 209.

It works. So x = -11 and we have |x +3| = |-11 + 3| = |-8| = 8.

For additional solutions to this problem such as completing the square, using the quadratic formula, creating a table of values, and by graphing check out New SAT Math Problems arranged by Topic and Difficulty Level.

Feel free to add your own solutions to the comments.

Comments

comments