For a positive integer , consider the system: The system has no real solution. Which choice is the greatest possible value of ?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 148 | Free SAT Prep | Aniko

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Question 148·200 Super-Hard SAT Math Questions·Advanced Math

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For a positive integer $k$ , consider the system:

\begin{cases} y = x^2 + 6{,}000 \\ y = kx + 675 \end{cases}

The system has no real solution. Cоnt‍еnt‌ by Anіkο.аi

Which choice is the greatest possible value of $k$ ?

When a system mixes a line and a nonlinear graph (like a parabola), substitute to turn it into a single quadratic equation in one variable. If the question is about whether real solutions exist, go straight to the discriminant: negative means no real intersection points. For a “greatest integer” parameter, turn the discriminant inequality into a bound like $k^2<\text{constant}$ and compare nearby perfect squares. A-⁠n-і-к-о.‍аі

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Eliminate a variable

Because both equations equal $y$ , set the right-hand sides equal to create a single equation in $x$ .

Think about when a quadratic has no real solutions

After substitution, you should get a quadratic in $x$ . Use the discriminant condition for having no real roots.

Estimate with nearby squares

You will need the largest integer $k$ such that $k^2$ is less than a fixed number. Compare the squares of consecutive integers. A‍niк⁠o.‌аi - SAT Р⁠rеp

Desmos Guide

Enter the parabola and the line with a slider

Enter

y = x^2 + 6000
y = kx + 675

When you type the second equation, create a slider for $k$ .

Look for values of $k$ with no intersections

Move the slider for $k$ upward. For each value, check whether the graphs intersect (Desmos shows intersection points when they exist). Anікo АІ Tuto‍r

Identify the greatest integer $k$ with no intersection

Find the largest integer setting of $k$ for which there are no intersection points, and confirm that increasing $k$ by 1 creates intersection point(s).

Step-by-step Explanation

Substitute to get one equation in one variable

Set the two expressions for $y$ equal:

x^2 + 6{,}000 = kx + 675

Rearrange to a quadratic in $x$ :

x^2 - kx + 5{,}325 = 0

Use the discriminant condition for no real solutions

\Delta=b^2-4ac<0

Here, $a=1$ , $b=-k$ , and $c=5{,}325$ , so

\Delta = (-k)^2 - 4(1)(5{,}325)=k^2-21{,}300

No real solution means

k^2-21{,}300<0 \quad\Rightarrow\quad k^2<21{,}300

Find the greatest integer $k$ that satisfies the inequality

Compare nearby perfect squares:

145^2=21{,}025

146^2=21{,}316

Since $21{,}025<21{,}300<21{,}316$ , the greatest integer $k$ with $k^2<21{,}300$ is 145.

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Step-by-step Explanation

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