Solve for the positive value of in the equation

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 7 | Free SAT Prep | Aniko

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Question 7·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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Solve for the positive value of $n$ in the equation

n^2 + 2n = 48

For quadratic equations like this on the SAT, first move everything to one side to get standard form $ax^2 + bx + c = 0$ . Then quickly check if the quadratic factors into two simple binomials by finding two numbers that multiply to $ac$ and add to $b$ . Use the zero product property to get the possible solutions, and finally apply any extra conditions in the question (such as "positive value") to choose the correct one from the answer choices.

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Get a standard quadratic form

Try to get all terms on one side so that the equation is equal to $0$ . What do you add or subtract from both sides to do this?

Factor the quadratic expression

Once you have $n^2 + 2n - 48 = 0$ , think of two numbers that multiply to $-48$ and add up to $2$ .

Use the factors to solve for n

After factoring, use the fact that if a product is $0$ , then at least one factor must be $0$ . This will give you two possible values for $n$ ; decide which one fits the condition in the question.

Desmos Guide

Graph the related quadratic

In Desmos, type y = x^2 + 2x - 48. This represents the equation after moving all terms to one side.

Find the x-intercepts

Look for the points where the graph crosses the x-axis (where $y = 0$ ). Note both x-values; these are the solutions to the equation, and the positive x-value is the one the question is asking for.

Step-by-step Explanation

Move all terms to one side

Start with the equation

n^2 + 2n = 48.

To solve a quadratic equation, first set it equal to $0$ by subtracting $48$ from both sides:

n^2 + 2n - 48 = 0.

Factor the quadratic

We want to factor $n^2 + 2n - 48$ into $(n + a)(n + b)$ .

Look for two numbers that:

multiply to $-48$ (the constant term), and
add to $2$ (the coefficient of $n$ ).

Those two numbers are $8$ and $-6$ , so the factored form is

(n + 8)(n - 6) = 0.

Use the zero product property and pick the positive solution

If a product of two factors is zero, then at least one factor must be zero. So set each factor equal to zero:

\begin{aligned} n + 8 &= 0 \ n - 6 &= 0 \end{aligned}

Solving these gives two values of $n$ , one negative and one positive. The problem asks for the positive value of $n$ , which is $6$ , so the correct answer choice is C.

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation