Solve any quadratic equation with the quadratic formula, and use the discriminant to determine the number and nature of the real solutions (Ohio A-REI.4b).

What this topic is asking

Ohio standard A-REI.4b asks you to solve any quadratic with the quadratic formula and to use the discriminant to count the real solutions. The formula is on the reference sheet, so the skill is substituting $a$, $b$, $c$ correctly and simplifying. The discriminant $b^2 - 4ac$, the part under the radical, tells you how many real roots there are. This is a high-value Quadratics skill, usable on either part.

Using the formula

The formula works on every quadratic, factorable or not, which makes it the reliable fallback.

The discriminant

The expression under the radical, $b^2 - 4ac$, controls how many real solutions exist, and you can read it before finishing.

A perfect-square discriminant also signals the quadratic factors over the integers, a useful cross-check.

Keeping signs and exact form

The most common errors are sign slips in $-b$ and in $-4ac$ when $a$ or $c$ is negative. Substitute each value in parentheses to keep signs. Unless the item says "round," leave the answer exact with the simplified radical, since exact-match items often expect the radical form.

How Ohio examines this topic

• Equation response. Enter the exact solutions from the math palette, including radicals.
• Multiple choice and multiple-select. Count real solutions from the discriminant, or pick the solution set.
• Numeric response. Compute the discriminant, or a decimal solution on the calculator part.

Why the discriminant counts the solutions

The number of real solutions hinges on the square root in the formula, and the discriminant is exactly what sits under that root. If $b^2 - 4ac$ is positive, its square root is a nonzero real number, and the $\pm$ then produces two different values, two real roots. If it is zero, the square root is $0$, so $+0$ and $-0$ give the same value, one repeated root. If it is negative, there is no real square root (you cannot take the real root of a negative), so there are no real solutions. This is why a quick look at the sign of $b^2 - 4ac$ settles the count before you do any further arithmetic, and why it matches the picture of how many times the parabola meets the $x$-axis.

Why the formula always works

The quadratic formula is what you get by completing the square on the general equation $ax^2 + bx + c = 0$ once and for all, with letters instead of numbers. Because that derivation makes no assumption about the particular coefficients, the resulting formula solves every quadratic, including those that do not factor with nice integers. That universality is why it is the dependable method: when factoring stalls or the roots are irrational, the formula still delivers the exact solutions. It also unifies the other methods, the square-root method and completing the square are special cases of the same algebra, which is why the formula and the discriminant carry so much of the Quadratics category.

Try this

Q1. Find the discriminant of $x^2 + 2x + 5 = 0$ and state the number of real solutions. [2 points]

• Cue. $2^2 - 4(1)(5) = 4 - 20 = -16 < 0$, so no real solutions.

Q2. Solve $x^2 - 6x + 9 = 0$ with the formula. [2 points]

• Cue. Discriminant $36 - 36 = 0$, so one root $x = \frac{6}{2} = 3$.