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Learn Quadratic Equations

Four ways to solve ax squared + bx + c = 0, and how to pick one.

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A quadratic equation has the form ax squared plus bx plus c equals 0, with a not zero. Its graph is a parabola, and its solutions are the points where that parabola crosses the x-axis. There are four standard methods for solving one: factoring, taking square roots, completing the square and the quadratic formula. Each has its moment, and choosing quickly is half the skill. The discriminant, b squared minus 4ac, tells you before solving whether you will get two real solutions, one or none. Learnr builds a focused quadratics course and teaches each method in short interactive lessons that check every line of your work.

How to learn Quadratic Equations, step by step

Get every equation into standard form. Move all terms to one side so the other side is 0 before solving. x squared = 5x - 6 becomes x squared - 5x + 6 = 0, and only then can you factor or apply the formula.
Factor when the numbers cooperate. Find two numbers that multiply to c and add to b: for x squared - 5x + 6, those are -2 and -3, giving (x - 2)(x - 3) = 0, so x = 2 or x = 3. Plug both back in to check.
Learn completing the square properly. For x squared + 6x + 5 = 0, move the 5, add 9 to both sides to get (x + 3) squared = 4, then x + 3 = 2 or -2, so x = -1 or x = -5. This method also produces vertex form.
Use the quadratic formula as the universal tool. x equals negative b, plus or minus the square root of b squared minus 4ac, all over 2a. Write a, b and c down first, including their signs, before substituting.
Read the discriminant and graph the parabola. A positive discriminant means two x-intercepts, zero means the vertex touches the axis, and negative means no real solutions. Sketching each case makes the algebra visible.
Practice choosing a method, a little every day. Mixed sets where you must decide which method fits are what build speed. Short daily Learnr lessons with a streak make that mixing easy to keep up.

What you'll learn

Solve quadratics by factoring and the zero product property
Solve by square roots and by completing the square
Apply the quadratic formula and simplify radicals
Use the discriminant to predict the number of real solutions
Graph parabolas from vertex form and find the vertex

What's inside your Quadratic Equations course

A sample path Learnr builds for Quadratic Equations. Your real course adapts to your goal and level.

1. Recognizing quadratics and standard form
2. Factoring and the zero product property
3. Square roots and completing the square
4. The quadratic formula and the discriminant
5. Graphing parabolas: vertex, axis and intercepts
6. Word problems: projectiles and area

How long does it take to learn Quadratic Equations?

With comfortable algebra skills, most learners can solve quadratics by every method in about 3 to 5 weeks at 3 to 4 hours a week. Factoring confidence is the usual bottleneck, and practicing it separately speeds everything else.

What you need before you start

Linear equations, multiplying binomials (FOIL), integer arithmetic and simplifying square roots. If factoring feels shaky, review it first.

Common mistakes to avoid

Dividing both sides by x, which throws away the solution x = 0. Factor out x instead and set each factor equal to zero.
Getting the sign of b wrong in the quadratic formula. If b is -5, then negative b is +5.
Writing the square root of 9 as only 3 when solving x squared = 9. There are two solutions, 3 and -3.

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Quadratic Equations questions, answered

Which method should I use to solve a quadratic?

Try factoring if the numbers are small and it looks clean. Use square roots when there is no x term. Otherwise, the quadratic formula always works, and completing the square is best when you need vertex form.

What does it mean when the discriminant is negative?

There are no real solutions, and the parabola never crosses the x-axis. The equation does have two complex solutions involving i, the square root of -1.