Solve quadratics by factoring, square roots, completing the square, and the quadratic formula.
Common CoreA-REI.B.4A-SSE.B.3A-APR.B.3
7 lessons25 short videos6 key concepts, out loud
15.1
Solving by Factoring (Zero-Product Property)
Set each factor to zero
A-REI.B.4A-APR.B.3
Key idea
The Zero-Product Property states that if the product of multiple factors is zero, then at least one factor must be zero, allowing you to solve quadratic equations by setting each linear factor to zero.
Learn how to solve quadratic equations by factoring them and setting each individual factor to zero.
Key concept · Lesson 15.1
Set it to zero, then split
Listen0:00 / 0:22
Transcript. A product can only be zero when one of its factors is zero. So write the equation with zero on one side, factor it, then set each factor equal to zero and solve those small equations separately. If the right side is anything but zero, this splitting is not allowed, so move everything across first.
15.2
Solving by Square Roots
Solve x^2 = k and (x - h)^2 = k
A-REI.B.4
Key idea
To solve quadratic equations of the form x^2 = k or (x - h)^2 = k, isolate the squared expression and take the square root of both sides, remembering to include both the positive and negative roots (±√k).
Learn how to solve quadratic equations by isolating the squared expression and taking the square root of both sides to find both solutions.
Key concept · Lesson 15.2
Isolate the square, then take both roots
Listen0:00 / 0:23
Transcript. Get the squared part completely alone before you take a square root. Squaring hides the sign, so when you undo a square you must write both roots, one positive and one negative. And if the right side ends up negative, stop. No real number squares to a negative, so there is no real solution.
15.3
Completing the Square
Make a perfect square to solve any quadratic
A-REI.B.4A-SSE.B.3
Key idea
To solve a quadratic equation by completing the square, add the square of half the linear coefficient, (b/2)^2, to both sides to create a perfect square trinomial that can be factored and solved using square roots.
Learn how to rewrite any quadratic equation as a perfect square binomial so you can solve it using square roots.
Key concept · Lesson 15.3
Add half of b, squared, to both sides
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Transcript. The magic number is half of the middle coefficient, squared. Halve it first, then square, or the number comes out far too big. Add that same number to both sides so the equation stays balanced, then factor the left side and finish with square roots.
15.4
The Quadratic Formula (Part 1: The Discriminant)
Predict how many solutions you'll get
A-REI.B.4
Key idea
The discriminant, b^2 - 4ac, is the value under the radical in the quadratic formula that determines whether a quadratic equation has two real solutions (if positive), one real solution (if zero), or zero real solutions (if negative).
Learn how to calculate the discriminant of a quadratic equation to predict how many solutions it has before you even start solving.
Key concept · Lesson 15.4
The sign predicts the root count
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Transcript. The discriminant is the part under the radical, b squared minus four a c. If it is positive you get two real solutions, if it is zero you get one, and if it is negative there are none. Always wrap a negative b in parentheses before squaring.
15.5
The Quadratic Formula (Part 2: Solving)
The formula that always works
A-REI.B.4
Key idea
To solve any quadratic equation in standard form ax^2 + bx + c = 0, substitute the coefficients a, b, and c into the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) and simplify the expression to find the exact solutions.
Learn how to use the quadratic formula to solve any quadratic equation, including those with radical or fractional solutions.
Key concept · Lesson 15.5
Whole numerator over 2a
Listen0:00 / 0:22
Transcript. Put the equation in standard form first, then read off a, b and c with their signs attached. Negative b flips the sign of b, so a negative b turns positive. The whole top, both terms, sits over two a, not just the radical part.
15.6
Choosing a Solution Method
Pick the best tool for the job
Key idea
To solve any quadratic equation efficiently, analyze its algebraic structure to select the fastest method: use square roots when b = 0, factoring when factors are easily found, completing the square when a = 1 and b is even, and the quadratic formula as the universal backup.
Learn how to analyze the structure of any quadratic equation to choose the fastest, most error-free solving method.
Key concept · Lesson 15.6
Let the structure pick the method
Listen0:00 / 0:22
Transcript. Check the structure before you start. With no middle term, take square roots. If the factors jump out, factor. If a is one and b is even, complete the square. Otherwise use the formula. It always works, but it costs the most arithmetic, so save it for last.
15.7
Review & Assessment
Mixed quadratic solving
Review, practice, quiz and test
Key idea
Review mixed quadratic solving with targeted practice, a practice test, and a chapter test.
This is one chapter of the climb.
Algebra 1 spans 16 chapters, every lesson taught on video with a key concept you can hear, and Vector on hand to answer questions out loud.
