Problem 60
Question
Solve the quadratic equation using any method. Find only real solutions. $$-3 x^{2}-2=7 x$$
Step-by-Step Solution
Verified Answer
The solutions of the given quadratic equation are \(x_{1}\) and \(x_{2}\), which can be found by substituting the values of \(a\), \(b\), and \(c\) into the quadratic formula
1Step 1: Rearranging the given equation
Rearrange the initial equation to transform it into the standard quadratic form. It means moving all terms to one side of the equation to get a zero on the other side. The equation given is \(-3x^{2} - 2 = 7x\). When rearranged, the equation becomes \(-3x^{2} - 7x - 2 = 0\).
2Step 2: Identify the coefficients
Identify the coefficients \(a\), \(b\), and \(c\) in the standard form \(ax^{2} + bx + c = 0\). In the equation \(-3x^{2} - 7x - 2 = 0\), \(a = -3\), \(b = -7\), and \(c = -2\).
3Step 3: Use the quadratic formula
Use the quadratic formula \(x = [-b \pm \sqrt(b^{2} - 4ac)] / 2a\) to find the solutions of the equation. By substituting \(a\), \(b\), and \(c\) into the formula, we get two solutions: \(x_{1} = [-(-7) + \sqrt{(-7)^{2} - 4*(-3)*(-2)}] / 2*(-3)\) and \(x_{2} = [-(-7) - \sqrt{(-7)^{2} - 4*(-3)*(-2)}] / 2*(-3)\)
Key Concepts
Quadratic FormulaCoefficients of a Quadratic EquationReal Solutions of Quadratic EquationsStandard Quadratic Form
Quadratic Formula
The quadratic formula is a critical tool used to solve equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients. To resolve any quadratic equation, you can apply the quadratic formula, \(x = [-b \pm \sqrt{b^2 - 4ac}] / 2a\). This formula derives from completing the square of the quadratic equation and provides the roots directly.
The formula includes a plus-minus symbol (\pm), indicating that it will give us two values of \(x\), one for the addition and one for the subtraction, hence it accounts for both possible solutions of a quadratic equation.
An important aspect when using the quadratic formula is to ensure that you've correctly identified each coefficient in order to apply the formula properly. Incorrect identification of coefficients can lead to wrong solutions.
The formula includes a plus-minus symbol (\pm), indicating that it will give us two values of \(x\), one for the addition and one for the subtraction, hence it accounts for both possible solutions of a quadratic equation.
An important aspect when using the quadratic formula is to ensure that you've correctly identified each coefficient in order to apply the formula properly. Incorrect identification of coefficients can lead to wrong solutions.
Coefficients of a Quadratic Equation
In algebra, the coefficients of a quadratic equation are the numerical or literal parts which multiply the variable \(x\). In a standard quadratic equation \(ax^2 + bx + c = 0\), the coefficient \(a\) is the number before \(x^2\), \(b\) is the number before \(x\), and \(c\) represents the constant term, with no \(x\) attached.
Understanding coefficients is essential for solving quadratic equations, as they are directly used in the quadratic formula. While identifying coefficients, it's crucial to pay attention to their signs, as they can be negative or positive and will impact the calculations when applying the quadratic formula.
Coefficients play a pivotal role in determining the nature and shape of a parabola represented by the quadratic equation on a graph. For example, \(a\) determines the direction the parabola opens, and if \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward.
Understanding coefficients is essential for solving quadratic equations, as they are directly used in the quadratic formula. While identifying coefficients, it's crucial to pay attention to their signs, as they can be negative or positive and will impact the calculations when applying the quadratic formula.
Coefficients play a pivotal role in determining the nature and shape of a parabola represented by the quadratic equation on a graph. For example, \(a\) determines the direction the parabola opens, and if \(a > 0\), the parabola opens upward, and if \(a < 0\), it opens downward.
Real Solutions of Quadratic Equations
When solving a quadratic equation, we might end up with two kinds of solutions: real or complex. Real solutions are the roots that can be plotted on a real number line, while complex solutions cannot, as they involve imaginary numbers.
The discriminant, \(b^2 - 4ac\), within the quadratic formula, is the key to determining the nature of the solutions. When the discriminant is positive, we get two distinct real solutions; if it's zero, there is one real solution (also referred to as a repeated or double root); and if the discriminant is negative, this indicates that the equation has two complex solutions.
For the quadratic equation \( -3x^2 - 7x - 2 = 0 \), the discriminant is \( (-7)^2 - 4(-3)(-2) = 49 - 24 = 25 \), a positive number, thus, we can be sure that it has two distinct real solutions.
The discriminant, \(b^2 - 4ac\), within the quadratic formula, is the key to determining the nature of the solutions. When the discriminant is positive, we get two distinct real solutions; if it's zero, there is one real solution (also referred to as a repeated or double root); and if the discriminant is negative, this indicates that the equation has two complex solutions.
For the quadratic equation \( -3x^2 - 7x - 2 = 0 \), the discriminant is \( (-7)^2 - 4(-3)(-2) = 49 - 24 = 25 \), a positive number, thus, we can be sure that it has two distinct real solutions.
Standard Quadratic Form
The standard quadratic form of an equation is given by \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are the real numbers and \(a eq 0\). It's vital for \(a\) to be non-zero because if \(a = 0\), the equation ceases to be quadratic and turns linear instead.
To solve a quadratic equation, first, ensure that it is in this standard form. Rearrange the terms by moving all of them to one side of the equation, leaving zero on the other side. This step is crucial for solving the equation correctly using methods like factoring, completing the square, or applying the quadratic formula.
The power of the standard form lies in its simplicity and uniformity which allows for straightforward application of solving techniques, and it's a form that is readily identified and relied upon in algebraic problem-solving.
To solve a quadratic equation, first, ensure that it is in this standard form. Rearrange the terms by moving all of them to one side of the equation, leaving zero on the other side. This step is crucial for solving the equation correctly using methods like factoring, completing the square, or applying the quadratic formula.
The power of the standard form lies in its simplicity and uniformity which allows for straightforward application of solving techniques, and it's a form that is readily identified and relied upon in algebraic problem-solving.
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