SAT Math: In 3x² - 12x + c = 0, If No Real Solutions, Find Least Integer Value of c

Exact Question Text

In the given equation, $c$ is a constant:

> $$ 3x^2 - 12x + c = 0 $$

If the equation has no real solutions, what is the least possible integer value of $c$?

Step-by-Step Algebraic Solution

1. Identify the Quadratic Coefficients:

In $ax^2 + bx + c = 0$: $a = 3$, $b = -12$, and constant term $= c$.

1. Apply the Discriminant Condition:

The number of real solutions is governed by $\Delta = b^2 - 4ac$:

$$ \Delta < 0 \implies \text{no real solutions} $$

1. Set up the Inequality:

$$ (-12)^2 - 4(3)(c) < 0 $$

$$ 144 - 12c < 0 \implies c > 12 $$

1. Determine the Least Integer:

Since $c > 12$, the smallest integer is $13$.

Fast Desmos 10-Second Shortcut

1. In Line 1, type: y = 3x^2 - 12x + c and add a slider for $c$.
2. At $c = 12$, the parabola touches the axis at $(2, 0)$.
3. For $c \ge 13$, the curve floats with 0 real roots. Answer: 13.