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

A equação quadrática ajuda a resolver problemas envolvendo área quando uma variável aparece como um termo quadrado.Considere um jardim retangular com um perímetro de 26 metros e uma área de 40 metros quadrados. O objetivo é encontrar suas dimensões.Seja um lado a variável, em metros, e expresse o outro lado em termos dessa variável usando a relação de perímetro.A substituição dessas expressões na fórmula da área resulta em um produto envolvendo uma única variável.Expandir e reorganizar essa expressão leva a uma equação quadrática.Um método para resolver essa equação quadrática é por fatoração, onde a equação é escrita como um produto de dois binômios, cada um definido como zero para encontrar os valores da variável.Outro método para resolver a equação é completar o quadrado. Isso envolve reescrever a equação adicionando uma constante a ambos os lados para formar um quadrado perfeito e tomando a raiz quadrada de ambos os lados.A fórmula quadrática é um terceiro método para resolver equações quadráticas. Envolve a substituição dos coeficientes da equação por uma fórmula padrão para encontrar os valores da variável.Todos os três métodos produzem as mesmas soluções, que satisfazem as condições do problema original.

equações quadráticas

Uma equa&#231;&#227;o quadr&#225;tica &#233; uma express&#227;o alg&#233;brica em que uma vari&#225;vel &#233; elevada ao quadrado e combinada com sua primeira pot&#234;ncia e uma constante, sendo todas elas igualadas a zero. Essas equa&#231;&#245;es s&#227;o amplamente utilizadas para modelar rela&#231;&#245;es que envolvem &#225;rea, movimento e otimiza&#231;&#227;o. A forma geral de uma equa&#231;&#227;o quadr&#225;tica &#233;<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>onde a, b e c s&#227;o valores reais, e a &#233; diferente de zero para garantir a presen&#231;a do termo quadr&#225;tico.Um dos m&#233;todos para resolver uma equa&#231;&#227;o quadr&#225;tica consiste em reescrever a equa&#231;&#227;o como o produto de duas express&#245;es lineares. Essa forma fatorada &#233; representada por<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>Nessa express&#227;o, r e s representam os valores que tornam cada fator igual a zero. De acordo com a propriedade do produto nulo, se o produto de duas express&#245;es &#233; igual a zero, ent&#227;o pelo menos uma delas deve ser zero. Dessa forma, os valores representados por r e s correspondem aos valores da vari&#225;vel que satisfazem a equa&#231;&#227;o. A substitui&#231;&#227;o de qualquer um desses valores na equa&#231;&#227;o resulta em uma afirma&#231;&#227;o verdadeira, confirmando-os como solu&#231;&#245;es.Quando a fatora&#231;&#227;o n&#227;o &#233; aplic&#225;vel, utiliza-se uma abordagem mais geral de resolu&#231;&#227;o por meio da f&#243;rmula padr&#227;o<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>Essa f&#243;rmula fornece todas as solu&#231;&#245;es poss&#237;veis da equa&#231;&#227;o, bastando substituir nela os valores provenientes da express&#227;o original. O termo dentro da raiz quadrada, denominado discriminante, determina a natureza das solu&#231;&#245;es. Um discriminante positivo resulta em duas solu&#231;&#245;es distintas, um discriminante igual a zero fornece uma solu&#231;&#227;o dupla, e um discriminante negativo indica que a equa&#231;&#227;o n&#227;o possui solu&#231;&#245;es reais. Esses m&#233;todos oferecem uma abordagem completa e sistem&#225;tica para a resolu&#231;&#227;o de qualquer equa&#231;&#227;o quadr&#225;tica.

Equações Quadráticas

Uma equa&#231;&#227;o quadr&#225;tica &#233; uma express&#227;o alg&#233;brica em que uma vari&#225;vel &#233; elevada ao quadrado e combinada com sua ...

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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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