Solve quadratic equations by inspection, square roots, factoring, and the quadratic formula, writing exact solutions (NC.M1.A-REI.4a).

What this topic is asking

NC.M1.A-REI.4a asks you to solve quadratic equations in one variable by inspection, taking square roots, factoring, and the quadratic formula, and to write solutions in exact form when appropriate. The solutions of $ax^2 + bx + c = 0$ are the zeros of the related function and the x-intercepts of its graph.

The zero-product property

Factoring solves a quadratic because of one key fact.

Solving by factoring

Solving by square roots

When there is no linear term, isolate $x^2$ and take square roots, keeping both signs.

For $3x^2 - 27 = 0$: add $27$ and divide by $3$ to get $x^2 = 9$, so $x = \pm 3$. The $\pm$ is essential; dropping it loses a solution.

The quadratic formula

When a quadratic does not factor nicely, the formula always works.

For $ax^2 + bx + c = 0$, $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Since NC Math 1 gives no reference sheet, memorize it. For $x^2 - 4x + 1 = 0$: $a = 1$, $b = -4$, $c = 1$, so $x = \dfrac{4 \pm \sqrt{16 - 4}}{2} = \dfrac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}$, an exact answer.

Choosing a method

• Square roots when there is no $x$ term ($x^2 = k$ or $a x^2 + c = 0$).
• Factoring when the trinomial factors over the integers.
• Quadratic formula when factoring is hard or the roots are irrational.

How the NC Math 1 EOC examines this topic

• Gridded response. Enter a solution (a zero) of a quadratic.
• Multiple choice. Choose the solution set, or which method is most efficient.
• Calculator-active. The quadratic formula and decimal estimates fit the calculator-active section, though factoring may appear in the calculator-inactive one.

Solving connects directly to factoring (the most common route), to key features (solutions are x-intercepts), and to the structure ideas in rewriting expressions.

Why a quadratic usually has two solutions

A parabola is symmetric, so it typically crosses the x-axis in two places, one on each side of its axis of symmetry. Those two crossings are the two solutions, which is why the quadratic formula carries a $\pm$ and why the square-root method keeps both signs. Sometimes the parabola just touches the axis (one repeated solution) or never reaches it (no real solution), but the default is two. Understanding this geometrically prevents the most common error, reporting only one root, and it links solving to graphing: solving $ax^2 + bx + c = 0$ is the same as finding where $y = ax^2 + bx + c$ meets the x-axis. The method you choose is only about convenience; all correct methods give the same zeros.

Try this

Q1. Solve $x^2 - 9x + 18 = 0$ by factoring. [2 points]

• Cue. $(x - 3)(x - 6) = 0 \Rightarrow x = 3$ or $x = 6$.

Q2. Solve $x^2 = 50$ in exact form. [1 point]

• Cue. $x = \pm\sqrt{50} = \pm 5\sqrt{2}$.