Name: solving quadratics by factoring and formula
Uploaded: 2025-11-26T05:44:30.000Z
Duration: 1 min 29 s
Channel: JoVE

Równanie kwadratowe pomaga rozwiązywać problemy związane z powierzchnią, gdy zmienna pojawia się jako składnik podniesiony do kwadratu.Rozważ prostokątny ogród o obwodzie 26 metrów i powierzchni 40 metrów kwadratowych. Celem jest odnalezienie jego wymiarów.Niech jedna strona będzie zmienną w metrach, a druga strona będzie wyrażona w kategoriach tej zmiennej za pomocą relacji obwodu.Podstawienie tych wyrażeń do formuły obszaru powoduje, że produkt obejmuje pojedynczą zmienną.Rozwinięcie i przegrupowanie tego wyrażenia prowadzi do równania kwadratowego.Jedną z metod rozwiązania tego równania kwadratowego jest rozkład na czynniki pierwsze, w którym równanie jest zapisywane jako iloczyn dwóch dwumianów, z których każdy jest ustawiony na zero, aby znaleźć wartości zmiennej.Inną metodą rozwiązania równania jest uzupełnienie kwadratu. Polega to na przepisaniu równania poprzez dodanie stałej po obu stronach, aby utworzyć idealny kwadrat i pobranie pierwiastka kwadratowego z obu stron.Wzór kwadratowy to trzecia metoda rozwiązywania równań kwadratowych. Polega na podstawieniu współczynników równania do standardowej formuły w celu znalezienia wartości zmiennej.Wszystkie trzy metody dają te same rozwiązania, które spełniają warunki pierwotnego problemu.

równania kwadratowe

R&#243;wnanie kwadratowe to wyra&#380;enie algebraiczne, w kt&#243;rym zmienna wyst&#281;puje w drugiej pot&#281;dze, a tak&#380;e w pierwszej pot&#281;dze oraz jako sta&#322;a; ca&#322;o&#347;&#263; jest r&#243;wna zeru. R&#243;wnania te cz&#281;sto s&#322;u&#380;&#261; do modelowania zale&#380;no&#347;ci dotycz&#261;cych pola, ruchu oraz optymalizacji. Og&#243;lna posta&#263; r&#243;wnania kwadratowego to<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:523/74;' src='https://cloudfront.jove.com/files/ftp_upload/40031/40031_Equation_1_editor_40031.svg' alt='40031_Equation_1.svg' /></figure>gdzie a, b i c s&#261; liczbami rzeczywistymi, a a jest r&#243;&#380;ne od zera, aby zapewni&#263; obecno&#347;&#263; cz&#322;onu kwadratowego.Jedn&#261; z metod rozwi&#261;zywania r&#243;wnania kwadratowego jest przekszta&#322;cenie go w posta&#263; iloczynu dw&#243;ch wyra&#380;e&#324; liniowych. T&#281; posta&#263; zapisuje si&#281; jako<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:514/74;' src='https://cloudfront.jove.com/files/ftp_upload/40031/40031_Equation_2_editor_40031.svg' alt='40031_Equation_2.svg' /></figure>W tym wyra&#380;eniu r i s oznaczaj&#261; warto&#347;ci, dla kt&#243;rych ka&#380;dy z czynnik&#243;w jest r&#243;wny zeru. Zgodnie z zasad&#261; iloczynu r&#243;wnego zeru, je&#380;eli iloczyn dw&#243;ch wyra&#380;e&#324; jest r&#243;wny zeru, to co najmniej jedno z nich musi by&#263; r&#243;wne zeru. W konsekwencji warto&#347;ci r i s s&#261; tymi warto&#347;ciami zmiennej, kt&#243;re spe&#322;niaj&#261; r&#243;wnanie. Podstawienie kt&#243;rejkolwiek z tych warto&#347;ci do r&#243;wnania daje zdanie prawdziwe, co potwierdza, &#380;e s&#261; to rozwi&#261;zania.Gdy rozk&#322;ad na czynniki nie jest mo&#380;liwy, stosuje si&#281; podej&#347;cie og&#243;lne z wykorzystaniem standardowego wzoru:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:50%;'><img style='aspect-ratio:649/187;' src='https://cloudfront.jove.com/files/ftp_upload/40031/40031_Equation_3_editor_40031.svg' alt='40031_Equation_3.svg' /></figure>Wz&#243;r ten pozwala wyznaczy&#263; wszystkie rozwi&#261;zania r&#243;wnania poprzez podstawienie warto&#347;ci z wyj&#347;ciowego wyra&#380;enia. Wyra&#380;enie pod znakiem pierwiastka, nazywane wyr&#243;&#380;nikiem, decyduje o charakterze rozwi&#261;za&#324;. Dodatni wyr&#243;&#380;nik prowadzi do dw&#243;ch r&#243;&#380;nych rozwi&#261;za&#324;, zerowy &#8212; do jednego rozwi&#261;zania podw&#243;jnego, natomiast ujemny informuje, &#380;e r&#243;wnanie nie ma rozwi&#261;za&#324; rzeczywistych. Metody te zapewniaj&#261; pe&#322;ne i systematyczne podej&#347;cie do rozwi&#261;zywania dowolnego r&#243;wnania kwadratowego.

Równania kwadratowe

R&#243;wnanie kwadratowe to wyra&#380;enie algebraiczne, w kt&#243;rym zmienna wyst&#281;puje w drugiej pot&#281;dze, a tak&#380;e w pierwszej ...

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Math Fundamentals

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Foundations of Mathematics

KEY TERMS AND CONCEPTS

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Quadratic Equations

solving quadratics by factoring and formula

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems.

The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions.

In factored form, the equation is written so that the factors contain values r and s. These values make each factor equal to zero. By the zero-product property, if a product equals zero, then at least one factor must be zero. That means r and s are the values of the variable that satisfy the equation, and substituting either value back into the original equation gives a true statement.

When factoring does not work, the quadratic formula gives a more general method. This formula finds all possible solutions by using the values from the original equation. The expression under the square root is called the discriminant. A positive discriminant gives two different real solutions, a zero discriminant gives one repeated solution, and a negative discriminant means there are no real solutions.

Solving Quadratics by Factoring and Formula

solving-quadratics-by-factoring-and-formula

real number system

Number Types on the Real Line

The real number system brings together the number types used for counting, measuring, and comparing quantities. It includes natural numbers, integers, rational numbers, and irrational numbers. Each set adds new kinds of numbers to the same system. Natural numbers are the basic counting numbers: 1, 2, 3, and so on. Integers extend that set by adding zero and negative whole numbers, such as ..., -3, -2, -1, 0, 1, 2, .... Rational numbers are numbers that can be written as a ratio of two...

real number operations

Real Numbers on the Number Line

Real numbers are the values that can be placed on a continuous number line. The system grew from counting numbers used for enumeration. It later expanded to include zero, which shows the absence of quantity, and negative numbers, which are opposites of counting numbers. The real number system also includes fractions and decimals that represent parts of a whole or equal divisions. When written in decimal form, these numbers may end or repeat in a regular pattern. Together, these values describe...

exponents

Exponent Rules and Scientific Notation

Exponents show repeated multiplication in a short form. They are important in algebra and in many other math topics, including scientific computation, scaling laws, and dimensional analysis. For a nonzero real number a and an integer n, a^n means a multiplied by itself n times. Exponent rules help simplify expressions with powers. These properties make it easier to work with exponential expressions in both symbolic and numerical form. They also support common calculations that use exponents in...

radicals

Simplifying Radicals

Radicals show the root of a number and the exponent needed to get that value. A radical expression has two main parts. The radicand is the number inside the root symbol, and the index tells which root to take. The notation n√a means the principal nth root of a. The index changes the type of root. When n equals 2, the expression is a square root. When n equals 3, it is a cube root. The sign of the radicand also matters. If the index is even, a negative radicand does not give a real result. If...

algebraic expressions

Simplifying Algebraic Expressions

Algebraic expressions use variables, constants, and operations to show mathematical relationships. They help describe patterns and solve problems across many areas of mathematics. Knowing the parts of an expression makes it easier to simplify and work with it correctly. Each algebraic expression has separate pieces that work together. A coefficient is the number attached to a variable, and an exponent shows repeated multiplication. Some expressions have a single part, while others include...

rational expressions

Working with Rational Expression Domains

Rational expressions are algebraic fractions with polynomials in the numerator and denominator. They follow the same basic rules as numerical fractions, but variables make them more delicate to work with. A key step is finding values that make the expression undefined, usually because of division by zero or an undefined radical. To determine the domain, start by identifying the values excluded from the denominator. The domain includes all real numbers except those that make the denominator...

linear equations

Solving Linear Equations

Linear equations are a basic algebra tool for modeling constant rates and everyday relationships. They use constants and a single variable, and each term is either a constant or a product of a constant and that variable. When graphed on a Cartesian coordinate plane, they form a straight line that shows a constant rate of change between two quantities. A linear equation in one variable often has the form ax + b = c, where a, b, and c are constants and x is the variable. Solving the equation...

quadratic equations

Quadratic equations are algebraic equations with a variable squared, a first-power term, and a constant term. They are set equal to zero. In high school math, they often help model area, motion, and optimization problems. The general form of a quadratic equation uses real numbers a, b, and c, with a not equal to zero. The nonzero a value keeps the squared term in the equation. One common way to solve a quadratic is to rewrite it as a product of two linear expressions. In factored form, the...

complex numbers

Complex Numbers in Solving Equations

Complex numbers extend the real number system so equations with negative square roots can still be solved. The real numbers cannot represent the square root of a negative number. That limits some quadratic equations, especially those with negative discriminants. To solve this problem, mathematicians introduced the imaginary unit i, where i = √(-1). A complex number is written as x + yi, where x and y are real numbers. In this form, x is the real part and y is the imaginary part. The key rule...

quadratic equations in the complex number system

quadratic-equations-in-the-complex-number-system

Complex Roots of Quadratic Equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

radical equations

Solving Radical Equations with Checks

Radical equations are equations with a variable inside a radical, most often a square root or cube root. These equations appear in science, engineering, and real-world measurement problems that involve nonlinear relationships. Solving them starts by isolating the radical expression. Then the radical is removed by raising both sides to the power that matches the radical’s index. This process can create extraneous solutions, which are values that work in the transformed equation but not in the...

inequalities

Solving Inequalities and Interval Notation

Inequalities show how two values compare when they are not equal. They use symbols such as , ≤, and ≥. These statements often describe a range of possible answers instead of one exact value. Interval notation is a shorter way to write these solution sets. An open interval, written as (a, b), leaves out the endpoints. A closed interval, written as [a, b], includes both endpoints. Half-open intervals, such as (a, b] and [a, b), include only one endpoint. Intervals can also extend forever in one...

absolute value inequalities

Absolute Value Inequalities on a Number Line

Absolute value inequalities show how far a number can be from zero or from a target value. Absolute value means distance on the number line, so the sign of the number does not matter. In inequalities, this distance helps define a valid range of values for a variable. An inequality of the form |x| ≤ a, where a ≥ 0, includes every x between −a and a. This can also be written as a compound inequality. On a number line, the solution is shown as a solid segment from −a to a with closed circles at...

mathematical modeling: problem solving

Mathematical Modeling for Loan Payments

Mathematical modeling turns a real-life situation into an equation that can be solved. It starts with defining the problem, choosing measurable quantities, and assigning variables to them. Then a matching model is selected and the equation is solved for the unknown value. This process is especially useful in finance, where models help evaluate investments, loans, and repayment plans. One common example is finding the fixed monthly payment on a loan. That calculation uses the standard annuity...

key terms and concepts

Learn how to solve quadratic equations by factoring or with the quadratic formula, and see how the discriminant shows whether there are real solutions.

Solving Quadratics by Factoring and Formula | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Solving Quadratics by Factoring and Formula

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