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Questions and Answers
Que représente le terme Δ dans la formule quadratique?
Quelle méthode suggère de résoudre un système d'équations en traçant les points sur un graphique?
Quelle méthode consiste à multiplier une équation par un scalaire pour éliminer une variable spécifique?
Comment résoudre une équation quadratique lorsque le discriminant est négatif?
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Quelle étape suit immédiatement l'obtention de x = -1/2 ± 5/2 lors de la résolution de l'équation quadratique?
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Comment résoudre une équation quadratique de la forme $ax^2 + bx + c = 0$?
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Quelle est la première étape pour résoudre une équation linéaire ax + b = c?
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Quelle méthode est utilisée pour résoudre une équation quadratique en réarrangeant l'expression en un trinôme carré parfait?
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Que signifie une équation linéaire de la forme 3x - 9 = 7 après avoir ajouté 9 des deux côtés?
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Quel est le résultat correct après avoir divisé les deux côtés de l'équation ⅔ x = 16 / 3 par 3?
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Study Notes
Solving Equations in Algebra: A Step-by-Step Guide
Solving algebraic equations is a fundamental skill used across various fields from finance to physics. In this guide, we'll explore how to tackle different types of equations using proven techniques.
Linear Equations in One Variable
Linear equations have the form ax + b = c, where a is the coefficient of the variable x, b represents the constant term, and c is another constant. To solve these equations, follow these steps:
- Isolate the variable by adding or subtracting terms on one side.
- Combine like terms.
- Set equal coefficients to zero if there is also a variable term.
- Solve for the remaining isolated variable.
- Check your solution(s) with the original equation.
For example, let's consider the linear equation 3x - 9 = 7. Follow these steps:
3x - 9 = 7
Add 9 to both sides: 3x - 9 + 9 = 7 + 9 => 3x = 16
Divide both sides by 3: â…” x = 16 / 3 => x = 8
Check our answer: 3 * 8 - 9 = 24 - 9 = 15
Since 15 equals 7, our solution is correct.
Quadratic Equations
Quadratic equations take the form ax^2 + bx + c = 0, where a, b, and c are constants. There are several methods to find their solutions, including:
- Factoring: If you can factor out two binomials, set each to zero and solve for
x. For instance,x^2 + 5x + 6 = (x+2)(x+3). - Completing the square: This method involves rearranging the expression into a perfect square trinomial or very close to it and taking its square root. Let's examine
x^2 + 5x + 6again. First, move five halves squared inside the parentheses and add and subtract five-halves squared outside: [ \begin{align*} x^2 + 5x + 6 &= (x + \frac{5}{2})^{2} - (\frac{5}{2})^2 \ &\Rightarrow (x+\frac{5}{2})^{2} - \frac{25}{4}\ \end{align*}] Now, take the square root of both sides: (x + \frac{5}{2} = ±\sqrt{\frac{25}{4}}), simplify to get (x = -\frac{1}{2} \pm \frac{5}{\sqrt{4}} = -\frac{1}{2} \pm \frac{5}{2}). - Using the quadratic formula: Calculate
Δ = b² - 4ac, and plug values back into the formula: [x=\frac{-b \pm \sqrt{Δ}}{2a}.] Note that complex numbers may arise when the discriminant (∆) is negative.
Systems of Linear Equations
Systems consist of multiple equations involving more than one unknown. Typically, we attempt to eliminate variables until reaching a single equation with only one variable left. Elimination methods include graphing, substitution, and elimination by addition or subtraction:
- Graphical Method: Plot points for each equation and observe their intersection point(s).
- Substitution Method: Solve for one variable in one equation, and substitute its value into the other equation.
- Elimination Method: Multiply one equation by a scalar to make the coefficients of a particular variable line up, so they cancel out upon addition/subtraction. Then, proceed to isolate the other variable.
By learning and applying these skills, you will become adept at solving increasingly challenging problems in algebra. As always, practice makes perfect!
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Description
Explore techniques for solving linear equations in one variable, quadratic equations, and systems of linear equations. Learn methods such as factoring, completing the square, and using the quadratic formula. Improve your problem-solving skills in algebra with this comprehensive guide.
