What is the zero-product property and why must the equation equal zero?

The zero-product property says that if a product of factors equals zero, then at least one factor is zero. That is why a quadratic must be written equal to zero before factoring, so you can set each factor to zero and solve. If the equation equals some other number, the factors do not split, because that property only works for zero.

Why do you need the plus-or-minus sign when taking square roots?

Because a positive number has two square roots, one positive and one negative, and both can solve the equation. Taking the square root as a solving step must allow both, so x squared equals 25 gives x equals plus or minus 5. Writing only the positive root loses the second solution.

How do you complete the square?

For x squared plus b x, add half of b, then squared, which is the quantity b over 2, squared. That makes a perfect-square trinomial equal to the quantity x plus b over 2, all squared. Add the same value to both sides of an equation, then finish by taking square roots, keeping the plus-or-minus.

What does the discriminant tell you?

The discriminant is b squared minus 4 a c, the expression under the radical in the quadratic formula. If it is positive there are two real solutions, if it is zero there is one repeated solution, and if it is negative there are no real solutions. The sign alone answers how many times the parabola crosses the x-axis.

How do you find the vertex and axis of symmetry of a parabola?

The axis of symmetry is x equals negative b over 2 a, which is not on the Ohio reference sheet, so memorize it. The vertex sits on the axis, so its x-coordinate is negative b over 2 a, and you get the y-coordinate by evaluating the function at that x. The sign of a tells you whether the vertex is a minimum, when a is positive, or a maximum, when a is negative.

When do you use the vertex versus the zeros in a quadratic model?

Use the vertex for greatest or least questions, such as maximum height, largest area, or minimum cost, because the vertex is the turning point. Use the zeros, found by solving the function equal to zero, for when the quantity is zero, such as when a projectile lands or a business breaks even. Then keep only solutions that make sense in context.

OhioMaths

Ohio Algebra I: a complete guide to quadratics

A deep-dive Ohio Algebra I guide to quadratics, a high-value block of the Algebra and Functions categories. Covers solving by factoring with the zero-product property, the square-root method and completing the square, the quadratic formula and discriminant, graphing parabolas and their key features, and modeling with quadratics.

Generated by Claude Opus 4.817 min readA-REI.4a, A-REI.4b, A-SSE.3a, F-IF.7a, F-IF.8a, A-CED.1, F-IF.4Updated 2026-06-13

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What this category demands
Solving by factoring
Square roots, completing the square, and the formula
Graphing parabolas
Modeling with quadratics
How this category is examined
Check your knowledge

What this category demands

This guide covers quadratics, a high-value Ohio Algebra I block spanning the Algebra (A-REI, A-SSE, A-CED) and Functions (F-IF) categories. It rewards three solving methods plus graphing and modeling. Each dot-point page has its own practice: solving by factoring, square roots and completing the square, the quadratic formula and discriminant, graphing quadratic functions, and quadratic applications.

Solving by factoring

Write the quadratic in standard form $= 0$ , factor, and apply the zero-product property: set each factor to zero. Never divide by $x$ (it loses the $x = 0$ root); factor it out instead. The solutions are the parabola's zeros.

Square roots, completing the square, and the formula

The square-root method solves $(\,)^2 = k$ as $\pm\sqrt{k}$ . Completing the square adds $\left(\frac{b}{2}\right)^2$ to make a perfect square and gives vertex form. The quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (on the reference sheet) solves any quadratic, and the discriminant $b^2 - 4ac$ counts the real roots: positive two, zero one, negative none.

Graphing parabolas

A quadratic graphs as a parabola: $a > 0$ opens up (vertex a minimum), $a < 0$ opens down (vertex a maximum). The axis of symmetry is $x = \frac{-b}{2a}$ (not on the reference sheet); the vertex is on it. Read zeros from factored form, the $y$ -intercept ( $c$ ) from standard form, and the vertex from vertex form $a(x - h)^2 + k$ .

Modeling with quadratics

In a model, the vertex answers maximum/minimum questions (greatest height, largest area) and the zeros answer when the quantity is zero (a projectile lands, break-even). Find the vertex input with $\frac{-b}{2a}$ , evaluate for the value, and discard solutions that make no sense (negative time or length).

How this category is examined

Equation and numeric entry. Solve by any method, enter exact roots, or find a vertex, intercept, or modeled value.
Graphing. Plot a parabola, or mark its vertex or zeros.
Multiple choice and multiple-select. Count solutions from the discriminant, read opening direction and max/min, or match a model.

Check your knowledge

Work these as you would for credit on the Ohio test.

Solve $x^2 - 5x + 6 = 0$ by factoring. (2 points)
Solve $x^2 = 10x$ . (2 points)
Solve $(x + 2)^2 = 25$ by square roots. (2 points)
What completes the square for $x^2 + 12x$ ? (1 point)
How many real solutions does $x^2 + x + 4 = 0$ have? (1 point)
Find the axis of symmetry and vertex of $f(x) = x^2 - 2x - 3$ . (2 points)
For $h(t) = -16t^2 + 64t + 5$ , find the maximum height. (2 points)

Solutions to check-your-knowledge questions

Work through each solution fully before reading on.

Step 1: Q1. Solve $x^2 - 5x + 6 = 0$ by factoring

Find two numbers that multiply to $6$ and add to $-5$ : those are $-2$ and $-3$ . Write the factored form and apply the zero-product property:

(x - 2)(x - 3) = 0 \Rightarrow x = 2 \text{ or } x = 3.

Step 2: Q2. Solve $x^2 = 10x$

Never divide both sides by $x$ , because that would lose the solution $x = 0$ . Instead, move all terms to one side and factor:

x^2 - 10x = 0 \Rightarrow x(x - 10) = 0 \Rightarrow x = 0 \text{ or } x = 10.

Step 3: Q3. Solve $(x + 2)^2 = 25$ by square roots

A perfect square set equal to a number is solved by taking the square root of both sides, keeping both the positive and negative roots:

x + 2 = \pm 5.

Subtract 2 from each: $x = 3$ or $x = -7$ .

Step 4: Q4. What value completes the square for $x^2 + 12x$ ?

The completing-the-square rule says to take half the coefficient of $x$ and then square it. Half of $12$ is $6$ , and $6$ squared is:

\left(\frac{12}{2}\right)^2 = 36.

Step 5: Q5. How many real solutions does $x^2 + x + 4 = 0$ have?

Compute the discriminant $b^2 - 4ac$ with $a = 1$ , $b = 1$ , $c = 4$ :

1 - 4(1)(4) = 1 - 16 = -15.

Because the discriminant is negative, the quadratic formula would require a square root of a negative number, so there are no real solutions.

Step 6: Q6. Find the axis of symmetry and vertex of $f(x) = x^2 - 2x - 3$

The axis of symmetry formula (not on the reference sheet, so memorize it) is $x = \frac{-b}{2a}$ . With $a = 1$ and $b = -2$ :

x = \frac{-(-2)}{2(1)} = 1.

Evaluate $f$ at $x = 1$ to get the vertex $y$ -coordinate: $f(1) = 1 - 2 - 3 = -4$ .

Final answer: axis of symmetry $x = 1$ , vertex $(1, -4)$ .

Step 7: Q7. Find the maximum height of $h(t) = -16t^2 + 64t + 5$

Since $a = -16 < 0$ the parabola opens downward, so the vertex is the maximum. Find the time of the peak:

t = \frac{-64}{2(-16)} = \frac{-64}{-32} = 2 \text{ seconds}.

Evaluate height at $t = 2$ : $h(2) = -16(4) + 64(2) + 5 = -64 + 128 + 5 = 69$ feet.

Final answer: maximum height $69$ feet at $t = 2$ seconds.

Sources & how we know this

Ohio's Learning Standards for Mathematics: Algebra 1 — Ohio Department of Education and Workforce (2024)
Algebra I course resources (blueprint, reference sheet, released items) — Ohio Department of Education and Workforce (2024)