The line is tangent to the parabola . Which of the following expresses all possible values of in terms of , , and ?

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SAT Nonlinear Equations & Systems Question 159 | Free SAT Prep | Aniko

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Question 159·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

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The line $y = mx + n$ is tangent to the parabola $y = x^2 + bx + c$ . Which of the following expresses all possible values of $b$ in terms of $m$ , $n$ , and $c$ ?

For line–parabola tangency problems, immediately translate the geometry into algebra: set the two equations equal to get a quadratic in x, then use the fact that a tangent line touches the parabola at exactly one point, meaning the quadratic has exactly one real solution. Apply the discriminant condition $D = 0$ , simplify carefully to an equation relating the parameters, and then solve for the requested variable, watching out for common mistakes like dropping the factor of 2 from $\sqrt{4k}$ or flipping signs inside the square root. This discriminant approach is usually faster and less error-prone than trying to find the point of tangency directly.

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Interpret the word 'tangent'

Think about what it means for a line to be tangent to a parabola in terms of intersection points. How many points do they share?

Turn the geometry into an equation

Set the expressions $y = mx + n$ and $y = x^2 + bx + c$ equal to each other and rearrange to get a quadratic equation in $x$ . What are the coefficients of this quadratic?

Use the discriminant for a quadratic

For the quadratic you found, write the discriminant in terms of $b$ , $m$ , $n$ , and $c$ . For tangency, how should this discriminant compare to zero, and what equation does that give you?

Desmos Guide

Write the discriminant in Desmos

In Desmos, type the expression for the discriminant of the intersection quadratic: (b - m)^2 - 4(c - n) on its own line. This represents the condition that determines tangency.

Test each answer choice in the discriminant

For each answer option, substitute its expression for b into the discriminant. For example, if an option says b = expression, type (expression - m)^2 - 4(c - n) as a new line in Desmos. Do this separately for all four options.

Identify the tangency condition

Look at how each substituted discriminant simplifies in Desmos. The option whose substituted discriminant simplifies to 0 (for all allowed values of m, n, and c where the square root is defined) is the one that correctly gives the condition for tangency.

Step-by-step Explanation

Set the equations equal to find intersection points

A point of intersection lies on both graphs, so their y-values must be equal:

mx + n = x^2 + bx + c.

Move all terms to one side to write a quadratic in $x$ :

x^2 + (b - m)x + (c - n) = 0.

The number of real solutions to this quadratic equals the number of intersection points between the line and the parabola.

Use the tangency condition (exactly one solution)

A line tangent to a parabola touches it at exactly one point, so the quadratic must have exactly one real solution.

For a quadratic $ax^2 + px + q = 0$ , this happens when the discriminant $D = p^2 - 4aq$ equals $0$ .

Here, $a = 1$ , $p = b - m$ , and $q = c - n$ , so the discriminant is

(b - m)^2 - 4(c - n).

Set this equal to $0$ for tangency:

(b - m)^2 - 4(c - n) = 0.

Isolate the squared term

Now solve the discriminant equation step by step.

Add $4(c - n)$ to both sides:

(b - m)^2 = 4(c - n).

This equation expresses a relationship between $b$ and the other parameters $m$ , $n$ , and $c$ .

Solve for b

To solve $(b - m)^2 = 4(c - n)$ for $b$ , take the square root of both sides. Remember that a square root can be positive or negative:

\sqrt{(b - m)^2} = \pm \sqrt{4(c - n)}.

|b - m| = 2\sqrt{c - n}

and therefore

b - m = \pm 2\sqrt{c - n}.

Add $m$ to both sides to get all possible values of $b$ :

\boxed{b = m \pm 2\sqrt{c - n}},

which matches choice A.

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation