Solve quadratic equations with the quadratic formula, and use the discriminant to determine the number and type of real solutions (A.PAR, Patterning and Algebraic Reasoning).

What this topic is asking

This Patterning and Algebraic Reasoning (A.PAR) standard covers the quadratic formula, the universal solver that works on every quadratic whether or not it factors, and the discriminant, the part under the radical that counts the real solutions without fully solving. On the Georgia Milestones EOC the quadratic formula is typically on the reference sheet, so the credit is for substituting correctly and simplifying to simplest radical form, and the discriminant is a frequent quick item. This is the safety-net method: if a quadratic resists factoring, the formula always finishes it.

Using the quadratic formula

Write the equation in standard form, 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, increasing the value under the radical.

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 matches 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 no real solutions. Computing just the discriminant is much faster than solving when the question only asks how many solutions there are.

How the Milestones examines this topic

• Numeric entry. Solve with the formula and enter the solutions in simplest radical form.
• 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.

Why the discriminant lives inside the formula

The discriminant is not a separate fact to memorize; it is the part of the quadratic formula under the square root, and that is exactly why it controls the solutions. If $b^2 - 4ac$ is positive, its square root is a real nonzero number, so the $\pm$ produces two distinct real solutions. If it is zero, the square root is zero and the $\pm$ adds and subtracts nothing, leaving one repeated solution, the double root where the vertex sits on the x-axis. If it is negative, the square root of a negative number is not real, so there are no real solutions and the parabola never reaches the axis. Seeing the discriminant as "the thing under the root that the $\pm$ acts on" means the count, the type, and the graph are all one computation, which is why an item can ask for the discriminant and the number of x-intercepts interchangeably.

Choosing the formula wisely

The quadratic formula always works, which makes it the reliable fallback, but it involves the most arithmetic, so it is not always the fastest. Reserve it for quadratics that do not factor with integers. A quick check helps: if no integer pair multiplies to $ac$ and adds to $b$, stop hunting for a factorization and go straight to the formula. Keep factoring for integer-factorable quadratics and the square-root property for equations with no linear term, where those methods are quicker. On the EOC, recognizing when to deploy the formula versus a faster method is part of working efficiently within the time limit.

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.