Solving quadratics by graphing means finding where the parabola y = ax² + bx + c crosses the x-axis. First get the equation in the form ax² + bx + c = 0. Then graph the function. Each x-intercept is a solution. A quadratic can have two, one or no real solutions.
Vertex form of a quadratic function is y = a(x − h)² + k, where (h, k) is the vertex. It shows the turning point of the parabola right away. For example, y = 2(x − 3)² + 1 has its vertex at (3, 1), opens up because a is positive, and has the axis of symmetry x = 3.
Graphing quadratic functions means drawing the U-shaped curve, called a parabola, for y = ax² + bx + c. Find the vertex with x = −b/(2a), draw the axis of symmetry through it, and plot the y-intercept and its mirror point. If a is positive the parabola opens up; if a is negative it opens down.
Rationalizing the denominator means rewriting a fraction so there is no radical in the denominator. For one square root, multiply the top and bottom by that root. For a sum like 3 + √2, multiply by its conjugate, 3 − √2. For example, 6/√3 = 6√3/3 = 2√3.
Multiplying radicals uses the product rule, √a · √b = √(ab). Multiply the numbers outside the radicals, multiply the numbers inside, and then simplify. For example, 2√3 · 5√6 = 10√18 = 30√2. For sums, use the distributive property or FOIL.
Adding and subtracting radicals works like combining like terms. You can only combine like radicals, which have the same number under the radical sign. Add or subtract the numbers in front, and keep the radical the same. For example, 3√5 + 4√5 = 7√5.
Simplifying square roots means rewriting a square root so no perfect square factor is left under the radical sign. Find the largest perfect square that divides the number, split it out, and take its square root. For example, √72 = √(36 · 2) = 6√2. This is called simplest radical form.
Factoring completely means writing a polynomial as a product of factors that can’t be factored any more. Take out the GCF first. Then use the number of terms to pick a method, and keep going until every factor is prime. For example, 2x² − 18 = 2(x + 3)(x − 3).
A perfect square trinomial is a trinomial that equals a binomial squared. It fits a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². The first and last terms are perfect squares, and the middle term is 2 times their roots. For example, x² + 10x + 25 = (x + 5)².
Factoring the difference of squares means rewriting a square minus a square as a sum times a difference. The rule is a² − b² = (a + b)(a − b). Find what is squared in each term, then write one plus and one minus. For example, x² − 25 = (x + 5)(x − 5). It is a special product run backward.