Math: 4661e2a9

ID: 4661e2a9

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Advanced Math)

Comment: Approach a nonlinear system of equations with the same tools you are used to.

Method 1: Because all the x values in the ordered pairs of the answer choices are unique, as soon as we figure out that part, we will have the correct answer. If we are looking for values of x that would satisfy the quadratic, then it makes sense to eliminate the y's.

x - y = 1

x + y = x^2 - 3

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x + x = x^2 - 3 + 1

2x = x^2 - 2

0 = x^2 - 2x - 2

At this point, you could either work through the quadratic formula or complete the square. I typically use the latter strategy, since it tends to be faster and also helps to derive the circle equation in certain circumstances. In short, push the "c" term of y = ax^2 + bx + c to the opposite side, cut the "b" term in half, square the result, and add to both sides.

2 = x^2 - 2x

2 + (-2/2)^2 = x^2 - 2x + (-2/2)^2

2 + 1 = x^2 - 2x + 1

3 = (x - 1)^2

√3 = x - 1

1 +/- √3 = x

The correct answer must be (A).

Method 2: Desmos shows two answers.

Because none of the answer choices has a negative x-value, you know you can focus on the point on the right: the correct answer must have an x-value equivalent to approximately 2.73205. Put the calculator to work. If you are unsure how to use your keyboard shortcuts to type the square root symbol, click on the "Show Keypad" icon in the bottom-left corner of the screen.

The correct answer must be (A). No other option presents a close call.

Method 3: Plug in the answer choices using a calculator. Only one combination of x- and y-values will work in both equations.

A. (1 + √3) - (√3) = 1 √ and (1 + √3) + (√3) = (1 + √3)^2 - 3 --> 1 + 2√3 = 1 + √3 + √3 + 3 - 3 -->

1 + 2√3 = 1 + 2√3 √

Answer choice (A) must be correct. However, for practice, you could disprove the others. Work with the easier equation first (in my mind, the top one).

B. (√3) - (-√3) = 1 --> √3 + √3 = 1 --> 2√3 = 1 X

C. (1 + √5) - (√5) = 1 √ and (1 + √5) + (√5) = (1 + √5)^2 - 3 --> 1 + 2√5 = 1 + √5 + √5 + 5 - 3 -->

1 + 2√5 = 1 + 2√5 + 2 --> 0 = 2 X

D. (√5) - (-1 + √5) = 1 --> √5 + 1 - √5 = 1 √ and (√5) + (-1 + √5) = (√5)^2 - 3 --> 2√5 - 1 = 5 - 3 -->

2√5 = 3 X