Solve quadratic equations using the quadratic formula, and use the discriminant to determine the number and nature of the real solutions (A.EI.6).

What this topic is asking

This part of A.EI.6 asks you to solve any quadratic with the quadratic formula and to use the discriminant to find the number and nature of the real solutions. On the Virginia Algebra I SOL these are Equations and Inequalities items: substitute into the formula and type the solutions, or determine how many real roots an equation has. The quadratic formula is on the formula sheet. They appear as fill-in-the-blank and multiple choice.

The quadratic formula

For a quadratic in standard form $ax^2 + bx + c = 0$ (with $a \ne 0$), the solutions are

This formula always works, factorable or not, which is why it is the universal method. The steps are mechanical but sign-sensitive: identify $a$, $b$, and $c$ with their signs, substitute, and simplify.

Note in step 2 how $-4(2)(-2) = +16$: the two negatives multiply to a positive, which is a frequent place to slip.

Reading the coefficients with signs

The most common formula errors come from signs. Write the equation in standard form first, then read off $a$, $b$, $c$ including any minus signs. For $x^2 - 7x + 10 = 0$, $b = -7$, so $-b = +7$. For $3x^2 + x - 4 = 0$, $a = 3$, $b = 1$, $c = -4$. Substituting a wrong sign throws off the whole calculation.

The discriminant

The expression under the square root, $b^2 - 4ac$, is the discriminant. Because it is what you take the square root of, its sign decides how many real solutions exist:

• $b^2 - 4ac > 0$: the root is a real positive number, and the $\pm$ gives two distinct real solutions.
• $b^2 - 4ac = 0$: the root is $0$, so $\pm 0$ gives a single value, one real solution (a double root).
• $b^2 - 4ac < 0$: the root is of a negative number, which is not real, so no real solutions.

You can answer "how many solutions?" by computing only the discriminant, without solving the whole equation.

Why the discriminant controls the number of roots

The discriminant governs the solution count because it lives inside the square root of the formula, and a square root behaves differently on positive numbers, zero, and negatives. If $b^2 - 4ac$ is positive, $\sqrt{b^2 - 4ac}$ is a real nonzero number, and adding versus subtracting it ($\pm$) produces two different values, two roots. If it is exactly zero, then $+\sqrt{0}$ and $-\sqrt{0}$ are both zero, so the two cases collapse into one value, a repeated (double) root, which graphs as a parabola just touching the $x$-axis at its vertex. If it is negative, there is no real square root, so the formula yields no real number, and the parabola never reaches the $x$-axis. So the discriminant is a one-number summary of how the parabola meets the $x$-axis: twice, once, or not at all. This is also why factoring sometimes fails: a quadratic factors nicely over the integers only when the discriminant is a perfect square.

How the SOL examines this topic

• Fill-in-the-blank. Solve with the formula and type both solutions, exact radicals included.
• Multiple choice. Use the discriminant to state the number of real solutions, or pick the solution set.
• Drag-and-drop. Order the substitution and simplification steps.

Try this

Q1. Solve $x^2 + 2x - 8 = 0$ with the formula. [2 points]

• Cue. $x = \frac{-2 \pm \sqrt{4 + 32}}{2} = \frac{-2 \pm 6}{2}$, so $x = 2$ or $x = -4$.

Q2. How many real solutions does a quadratic with discriminant $0$ have? [1 point]

• Cue. One (a double root).