What are the four ways to solve a quadratic equation?

Factoring, the quadratic formula, completing the square, and taking square roots directly (for equations like x^2 = 49 with no x term). Graphing gives a visual estimate but is not exact. The three main algebraic methods are covered step by step above.

How do I know which method to use?

Try factoring first if the coefficients are small integers — it is the fastest when it works. If nothing factors within a few seconds, switch to the quadratic formula, which always works. Use completing the square when the problem asks for it or when you also need vertex form.

What does the discriminant actually tell you?

The discriminant b^2 - 4ac predicts the roots before you solve: positive means two distinct real roots, zero means exactly one repeated root, and negative means two complex roots. It is also a quick factoring test — if it is a perfect square, the quadratic factors over the integers.

Can a quadratic equation have no solution?

It can have no real solutions — that happens when the discriminant is negative and the parabola never crosses the x-axis. But it always has exactly two solutions if you allow complex numbers, counted with multiplicity.

Does Photo Math Solver handle quadratics that do not factor?

Yes. It tries factoring first since that gives the cleanest steps, and automatically falls back to the quadratic formula when the expression does not factor over integers — including irrational and complex roots, shown with full working.

You can start solving quadratic equations for free by adding the Chrome extension — try it on your own homework before deciding if you need anything more.

Can it read handwritten or textbook problems?

It reads anything you can screenshot on your screen: textbook pages, PDF worksheets, online quizzes, and photos of printed or clearly handwritten problems. Superscripts, fractions, and square roots are recognized as real math, not plain text.

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Photo Math Solver Solved

x^2 - 5x + 6 = 0

1Look for two numbers that multiply to $6$ and add to $-5$ : they are $-2$ and $-3$ .
2Factor: $(x - 2)(x - 3) = 0$ .
3Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$ .
4Solve: $x = 2$ or $x = 3$ .

On this page01Definition 02Formulas 03Methods 04Examples 05Practice 06Mistakes 07FAQ

01Definition

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two — the highest power of the variable is a square. In standard form it is written $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are numbers and $a \neq 0$ . If $a$ were zero, the $x^2$ term would vanish and the equation would just be linear.

Graphically, the left side $y = ax^2 + bx + c$ draws a parabola, and solving the equation means finding where that parabola crosses the x-axis. That is why a quadratic equation has at most two real solutions (called roots or zeros): a parabola can cross the axis twice, touch it once, or miss it entirely.

Which of those three cases you are in is decided by the discriminant, $b^2 - 4ac$ . Positive means two real roots, zero means one repeated root, and negative means the parabola never reaches the axis — the two roots are complex numbers instead.

Where you'll actually use this

Projectile motion: the height of a thrown ball over time is a quadratic, and its roots tell you when it lands.
Business: revenue and profit curves are often quadratic, and their roots mark break-even points.
Geometry: finding a rectangle’s dimensions from its area and perimeter leads directly to a quadratic.
Physics: stopping-distance and free-fall formulas are quadratic in time and speed.

Root (zero): A value of $x$ that makes the equation true — where the parabola crosses the x-axis.
Discriminant: The quantity $b^2 - 4ac$ under the square root in the quadratic formula; its sign predicts the number and type of roots.
Vertex: The highest or lowest point of the parabola, at $x = -\frac{b}{2a}$ — exactly halfway between the two roots.
Coefficient: The numbers $a$ , $b$ , $c$ multiplying each power of $x$ in standard form.

02Formulas

The formulas you need

Keep these on hand — every method below builds on them.

Standard form

ax^2 + bx + c = 0,\quad a \neq 0

Always rearrange into this form first — every method below assumes it.

Quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Works on every quadratic, whether or not it factors.

Discriminant

\Delta = b^2 - 4ac

Δ > 0: two real roots · Δ = 0: one repeated root · Δ < 0: two complex roots.

Factored form

a(x - r_1)(x - r_2) = 0

The roots r₁ and r₂ multiply to c/a and add up to −b/a (Vieta’s formulas).

03Methods

How to solve quadratic equations

Pick the method that fits the problem in front of you.

When to use it: Fastest when the coefficients are small integers and the equation factors cleanly — typical for most homework problems. Try it first.

1Put the equation in standard form: $ax^2 + bx + c = 0$ .
2For $a = 1$ : find two numbers that multiply to $c$ and add to $b$ .
3Rewrite as a product of two binomials: $(x - r_1)(x - r_2) = 0$ .
4Set each factor equal to zero and solve the two mini-equations.
5Check both answers by substituting them back into the original equation.

04Worked examples

Quadratic Equations, solved step by step

From easy to hard — pick a problem to see its full solution.

x^2 - 5x + 6 = 0

1Look for two numbers that multiply to $6$ and add to $-5$ : they are $-2$ and $-3$ .
2Factor: $(x - 2)(x - 3) = 0$ .
3Set each factor to zero: $x - 2 = 0$ or $x - 3 = 0$ .
4Solve: $x = 2$ or $x = 3$ .

Try it on your own quadratic equations homework

Photo Math Solver reads a quadratic equation directly from a screenshot — a textbook page, a PDF worksheet, an online quiz — and returns the full solution: the method it used, every intermediate step, and both roots. No retyping ax² + bx + c into a calculator, and no guessing whether it factors.

05Practice

Now you try

Work each one on paper first, then check your answer.

Problem 1

x^2 - 7x + 12 = 0

Problem 2

x^2 - 4x - 5 = 0

Problem 3

3x^2 - 5x + 2 = 0

Problem 4

x^2 + 4x + 1 = 0

Stuck on your quadratic equations homework? Screenshot it

Quadratics with messy coefficients or fractions are easy to mistype into a calculator. Screenshot the equation with Photo Math Solver wherever you find it — a PDF worksheet, a textbook photo, an online quiz — and get every step without re-entering the numbers by hand.

06Common mistakes

Where points get lost

Each of these shows up on real graded work — and each has a simple fix.

The mistake	Why it happens	The fix
Dropping the $\pm$ and reporting only one root.	Calculators and mental shortcuts return a single square root, so the second solution silently disappears.	Every time you take a square root while solving, write $\pm$ immediately — before simplifying anything else — and carry both branches to the end.
Mis-computing the discriminant when $b$ or $c$ is negative.	The double negative in $-4ac$ trips people up: for $c = -2$ , the term $-4ac$ becomes positive, which feels wrong when rushing.	Write the substitution with explicit parentheses first — $\Delta = (3)^2 - 4(2)(-2)$ — and only then evaluate the signs.
Dividing both sides by $x$ in equations like $x^2 = 5x$ .	Dividing by $x$ looks like legitimate simplification, but it quietly assumes $x \neq 0$ and destroys one of the two roots.	Move everything to one side and factor instead: $x^2 - 5x = 0$ gives $x(x - 5) = 0$ , so $x = 0$ or $x = 5$ .
Applying the formula before the equation is in standard form.	Problems are often given as $2x^2 = 3 - x$ , and it is tempting to read $a$ , $b$ , $c$ straight off the page as written.	Rearrange to $2x^2 + x - 3 = 0$ first, and only then identify $a = 2$ , $b = 1$ , $c = -3$ .
Treating a negative discriminant as "no solution."	Many courses first teach "no real solution," and the shorter phrase "no solution" sticks.	Say "no real solutions" and, if your course covers complex numbers, finish the problem: $\sqrt{-16} = 4i$ leads to a perfectly valid pair of complex roots.

Photo Math Solver shows every intermediate step, so slips like these are easy to catch before they cost you marks.

07FAQ