Quadratic Equations

A quadratic equation is any equation that rearranges to with . The presence of that term changes everything: where a linear equation has exactly one solution, a quadratic usually has two, and the methods for finding them are different from anything in the Linear Equations lesson.

Geometrically, the two solutions are the two places where the parabola crosses the -axis. A parabola can cross the axis twice (two real solutions), just touch it once (one repeated solution), or miss it entirely (two complex solutions). Every method in this lesson is a different way of finding those crossing points algebraically.

There are four standard methods. This lesson covers the two quickest — factoring and the square-root property — and gives you a decision guide for all four. Completing the Square and the Quadratic Formula each get their own lesson because they carry more machinery. Whichever method you use, the first job is almost always the same: get the equation into standard form with zero on one side, combining like terms, distributing parentheses, and clearing fractions with the LCD along the way.

The Quadratic Equation Formula

is the leading coefficient, is the coefficient of the linear term, and is the constant. Every method starts by getting the equation into this form, with zero on one side — except the square-root property, which isolates the squared expression instead.

Method 1: Factoring and the Zero-Product Property

The zero-product property: if , then or .

To use it, get one side equal to zero, factor the other side, set each factor to zero, and solve each small equation.

The factoring itself is the Factoring Quadratics skill; the new part here is the zero-product step that turns a factored quadratic into two linear equations.

Why the property holds: if two numbers multiply to zero, at least one of them must be zero — there is no other way for a product to vanish. So once the quadratic is written as a product of two factors equal to zero, the only way the equation can be true is for one factor or the other to be zero, and each of those is a small linear equation you already know how to solve. This is exactly why one side has to be zero before you factor. tells you nothing directly, because a product of can be made many ways; you have to expand, move the over, and re-factor first.

A repeated root

If the quadratic is a perfect square, both factors are the same and the two solutions coincide.

The parabola for a perfect-square quadratic just touches the -axis at one point instead of crossing it. The solution is still counted as “two equal roots” for bookkeeping, but there is only one distinct value.

The no-constant-term case

If , there is nothing to factor into a pair of binomials — instead, factor out of both terms. One of the roots is always .

Never divide both sides by to “simplify” this. Dividing by assumes and silently discards the solution.

Method 2: The Square-Root Property

When there is no term, isolate the square and take the square root of both sides — with a , because two numbers square to the same value.

The is not optional. Both and square to , so has two solutions, and . Writing only is the single most common error with this method. When you take the square root of a squared binomial, the attaches to the number on the other side, then you solve the resulting pair of linear equations for .

If , the solutions are complex: gives , and gives , so .

When to reach for the square-root property

Use it whenever the equation has no plain term — it is far faster than factoring or the formula in that case. It also applies the moment an equation is already a squared expression equal to a number, which is exactly the state completing the square puts every quadratic into. In other words, completing the square is just a way of forcing any quadratic into square-root-property form.

Method 3 and Method 4

• Completing the square rewrites as , turning any quadratic into a square-root-property problem. Full treatment in Completing the Square.
• The quadratic formula, , is completing the square done once, in general. It solves every quadratic. Full treatment in The Quadratic Formula.

Choosing a Method

Worked Example A: Factoring

Solve .

Step 1 — factor: two numbers multiplying to and adding to are and .

Step 2 — zero-product property:

Answer: or .

Worked Example B: Factoring with a GCF and a Leading Coefficient

Solve .

Step 1 — factor out the GCF :

Step 2 — the constant factor is never zero, so set the variable factors to zero:

Answer: or .

Worked Example C: The Square-Root Property

Solve .

Step 1 — isolate the square:

Step 2 — take the square root of both sides, with :

Step 3 — solve both:

Answer: or .

Worked Example D: Rearranging First

Solve .

Step 1 — move every term to one side so the equation equals zero:

Step 2 — factor and apply the zero-product property:

Answer: or . The first move — getting to standard form — is essential; the zero-product property is meaningless if the right side is not zero.

Worked Example E: A No-Constant-Term Quadratic

Solve .

Step 1 — move every term to one side:

Step 2 — factor out the greatest common factor, :

Step 3 — the constant is never zero, so set each variable factor to zero:

Answer: or . Dividing the original equation by would have given only and lost .

Worked Example F: A Quadratic Hiding Behind Parentheses

Solve .

Step 1 — expand the left side:

Step 2 — move the over so one side is zero:

Step 3 — factor and apply the zero-product property:

Answer: or . The tempting shortcut — setting and — is wrong, because the product on the left equals , not ; the zero-product property only applies to a product equal to zero.

Common Mistakes to Avoid

• Applying the zero-product property when one side isn’t zero. does not give ; expand and set equal to zero first.
• Dropping a solution. A quadratic has two roots. Reporting only when also works is the most common error.
• Forgetting the with the square-root property. has solutions and .
• Dividing both sides by . In , dividing by loses the solution . Factor instead: .
• Only taking the square root of one term. becomes , not with the forgotten, and not .
• Not factoring out a GCF first. is much easier as .

Where This Shows Up Later

• Completing the Square and the Quadratic Formula are the methods for the quadratics that do not factor.
• Quadratic Applications. Area, projectile, and product word problems all end in a quadratic equation solved by these methods.
• Polynomial Inequalities. Solving starts by finding the roots of .
• Graphing parabolas. The solutions of are the -intercepts of the parabola .
• Equations Reducible to Quadratic Form. Equations like become quadratics after a substitution.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Quick Reference

The factoring itself is Factoring Quadratics. For quadratics that don’t factor, continue to Completing the Square and The Quadratic Formula. Word problems that end in a quadratic are in Quadratic Applications. More Algebra lessons are available too.