Problem 99
Question
Exercises \(98-100\) will help you prepare for the material covered in the first section of the next chapter. In Exercises \(98-99,\) solve each quadratic equation by the method of your choice. $$ -x^{2}-2 x+1=0 $$
Step-by-Step Solution
Verified Answer
The solution to the quadratic equation \(-x² - 2x + 1 = 0\) are \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\).
1Step 1: Identify the coefficients
The given quadratic equation is \(-x^{2}-2x+1=0\). For a quadratic equation in the form \(ax^{2} + bx + c = 0\), \(a\), \(b\) and \(c\) are the coefficients. Here, \(a = -1\), \(b = -2\) and \(c = 1\).
2Step 2: Apply the Quadratic Formula
The roots of a quadratic equation \(ax^{2} + bx + c = 0\) can be given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]. Substitute \(a = -1\), \(b = -2\) and \(c = 1\) into the quadratic formula.
3Step 3: Substitute and simplify
The substituting gives: \[x = \frac{-(-2) \pm \sqrt{(-2)^{2}- 4(-1)(1)}}{2(-1)}\] This simplifies to: \[ x = \frac{2 \pm \sqrt{4 + 4}}{-2}\]. Finally simplifying we get: \[x = \frac{2 \pm \sqrt{8}}{-2}\] This further simplifies to: \[x = -1 \pm \sqrt{2}\] Hence, the roots are \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\).
4Step 4: Verification
To verify the roots, substitute them back into the original equation and check if both sides of the equation balance. On substituting each root, the Left Hand Side (LHS) should equal the Right Hand (RHS) Side, confirming that they are indeed the roots.
Key Concepts
Quadratic FormulaRoots of a Quadratic EquationQuadratic Equation Coefficients
Quadratic Formula
When faced with a quadratic equation in the form of \(ax^{2} + bx + c = 0\), the go-to method for finding solutions is the quadratic formula. This powerful tool offers a reliable way to find the roots, or solutions, no matter how tricky the coefficients may seem.
The formula states that the solutions for \(x\) can be found using: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]. The ease of this formula lies in its clear structure: identify the coefficients \(a\), \(b\), and \(c\), plug them into their respective places, and perform the arithmetic. Whether the coefficients are positive, negative, integers, or fractions, the quadratic formula navigates through all those complexities to reach the solution.
For the quadratic equation \(-x^{2}-2x+1=0\), the application of the quadratic formula reveals the roots to be \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\), illustrating the formula's capacity to tackle negative coefficients and provide exact answers for roots.
The formula states that the solutions for \(x\) can be found using: \[x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\]. The ease of this formula lies in its clear structure: identify the coefficients \(a\), \(b\), and \(c\), plug them into their respective places, and perform the arithmetic. Whether the coefficients are positive, negative, integers, or fractions, the quadratic formula navigates through all those complexities to reach the solution.
For the quadratic equation \(-x^{2}-2x+1=0\), the application of the quadratic formula reveals the roots to be \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\), illustrating the formula's capacity to tackle negative coefficients and provide exact answers for roots.
Roots of a Quadratic Equation
The roots of a quadratic equation represent the values of \(x\) that satisfy the equation \(ax^{2} + bx + c = 0\) by making it equal to zero. These roots are essentially the solution to the equation and can be thought of as the 'X' marks that solve the algebraic puzzle. Depending on the discriminant \(b^{2} - 4ac\), roots can be real or complex and are usually expressed as two values due to the \(\pm\) sign in the quadratic formula.
In our example, \(-x^{2}-2x+1=0\), the roots \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\) are both real. They can be verified by substituting into the original equation; if the equation balances out giving zero, this confirms their validity. Understanding roots is crucial not only in algebra but also in graphing quadratics, where roots represent the points where the parabola intersects the x-axis—another vital interpretation of the same mathematical principle.
In our example, \(-x^{2}-2x+1=0\), the roots \(x = -1 + \sqrt{2}\) and \(x = -1 - \sqrt{2}\) are both real. They can be verified by substituting into the original equation; if the equation balances out giving zero, this confirms their validity. Understanding roots is crucial not only in algebra but also in graphing quadratics, where roots represent the points where the parabola intersects the x-axis—another vital interpretation of the same mathematical principle.
Quadratic Equation Coefficients
The coefficients in a quadratic equation are the numerical multipliers of the terms, labelled \(a\), \(b\), and \(c\) in the general form \(ax^{2} + bx + c = 0\). These numbers shape the parabola's width, direction, and position on a coordinate plane. More specifically, \(a\) affects the opening direction and width, \(b\) influences the axis of symmetry, and \(c\) dictates where the graph intersects the y-axis.
For the example at hand, the coefficients \(a = -1\), \(b = -2\), and \(c = 1\) suggest a downward-opening parabola due to the negative \(a\), and a vertex not centered on the y-axis given by a nonzero \(b\). Recognizing the role of each coefficient is fundamental in understanding quadratic equations and opens the door to a more profound study of their graphs. It's akin to knowing the individual ingredients in a recipe that, when combined, create a distinctive dish, or in our case, a unique parabola.
For the example at hand, the coefficients \(a = -1\), \(b = -2\), and \(c = 1\) suggest a downward-opening parabola due to the negative \(a\), and a vertex not centered on the y-axis given by a nonzero \(b\). Recognizing the role of each coefficient is fundamental in understanding quadratic equations and opens the door to a more profound study of their graphs. It's akin to knowing the individual ingredients in a recipe that, when combined, create a distinctive dish, or in our case, a unique parabola.
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