Solve quadratic equations by applying the quadratic formula, and use the discriminant to determine the number of real solutions (TN A1.A.REI.B.4).

What this topic is asking

The quadratic formula is the universal solver in A1.A.REI.B.4: it works on every quadratic, factorable or not, and it is printed on the Math EOC reference sheet, so the credit is for substituting correctly and simplifying. The discriminant (the part under the radical) tells you the number of real solutions without fully solving, a frequent quick item.

Using the quadratic formula

Set the equation to $ax^2 + bx + c = 0$, identify $a$, $b$, $c$, and substitute into the formula.

The two error-prone spots are the sign of $-b$ and the sign of $-4ac$. When $c$ is negative, $-4ac$ becomes positive, which increases the discriminant.

The discriminant: counting real solutions

The discriminant is $b^2 - 4ac$, the expression under the radical. Its sign tells you how many real solutions exist, which corresponds to how many times the parabola meets the $x$-axis.

• $b^2 - 4ac > 0$: two real solutions (the parabola crosses the $x$-axis twice).
• $b^2 - 4ac = 0$: one real solution, a double root (the vertex touches the $x$-axis).
• $b^2 - 4ac < 0$: no real solutions (the parabola misses the $x$-axis).

For $x^2 + 2x + 5 = 0$: $b^2 - 4ac = 4 - 20 = -16 < 0$, so there are no real solutions. Computing just the discriminant is much faster than solving when the question only asks how many solutions there are.

How TNReady examines this topic

• Equation response. Solve with the formula and enter the solutions in simplest radical form; exact-match scoring rewards the reduced radical.
• Multiple choice. Count real solutions from the discriminant, with sign-error distractors.
• Inline choice. State the number of solutions and whether the parabola crosses the axis.

A clarifying idea is that the discriminant lives inside the formula, so the formula and the count are the same computation: if the discriminant is negative, the square root of a negative number has no real value, which is exactly why there are no real solutions.

Why the quadratic formula always works

The quadratic formula is derived by completing the square on the general equation $ax^2 + bx + c = 0$, which is why it solves every quadratic, including those that do not factor with integers. That universality makes it the safe fallback: if a quick check shows no integer factor pair multiplies to $ac$ and adds to $b$, go straight to the formula rather than hunting for a factorization that does not exist. The trade-off is more arithmetic, so reserve it for the quadratics that resist factoring, and keep factoring or square roots for the cases where they are faster. On the calculator subparts you may evaluate the formula numerically, but exact-match items still expect simplest radical form, so reduce $\sqrt{40}$ to $2\sqrt{10}$ rather than entering a decimal unless the item asks for a rounded value.

Connecting the discriminant to the graph

The discriminant has a clean graphical meaning that the test likes to probe. Because the solutions are the $x$-intercepts of $y = ax^2 + bx + c$, a positive discriminant means the parabola crosses the $x$-axis at two distinct points, a zero discriminant means it touches the axis at exactly one point (the vertex sits on the axis), and a negative discriminant means the parabola never reaches the axis. Reading the discriminant therefore tells you the shape of the picture, not just a count.

Try this

Q1. Solve $x^2 + 4x + 1 = 0$ in simplest radical form. [2 points]

• Cue. $x = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3}$.

Q2. How many real solutions does $x^2 - 6x + 9 = 0$ have? [1 point]

• Cue. $b^2 - 4ac = 36 - 36 = 0$, so one (double) solution.