Problem 27 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$-y 2+y+1=0$$

Step-by-Step Solution

Answer

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Step 1: Write the Equation

We need to solve: \(-y 2+y+1=0\).

Step 2: Isolate the Variable

We apply algebraic operations to isolate the variable. This may include adding, subtracting, multiplying, or dividing both sides by the same value.

Step 3: Solve for the Variable

After performing the algebraic manipulations, we find the solution.

Step 4: State the Solution

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Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. These equations are in the form of \( ay^2 + by + c = 0 \), where \( a \), \( b \), and \( c \) are constants known as coefficients. To find the roots of a quadratic equation, use the quadratic formula:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps determine the values of \( y \) that satisfy the equation. The expression under the square root \( b^2 - 4ac \) is called the discriminant. It is crucial in determining the nature of the roots.

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, called a double root.
• If it is negative, the roots are complex and occur in conjugate pairs.

Understanding the quadratic formula is essential for solving these types of equations effectively.

Coefficients in Quadratic Equations

In the context of quadratic equations, coefficients determine the shape and position of the parabola. A standard quadratic equation is written in the form \( ay^2 + by + c = 0 \). Let's delve into what these coefficients represent:- **\( a \)**: This coefficient affects the parabola's direction and width. If \( a > 0 \), the parabola opens upwards; if \( a < 0 \), it opens downwards. The magnitude of \( a \) influences how "stretched" or "compressed" the parabola looks.- **\( b \)**: This coefficient affects the parabola's symmetry. It helps calculate the axis of symmetry, given by the formula \(-\frac{b}{2a}\).- **\( c \)**: This constant term represents the parabola's y-intercept, the point where the graph crosses the y-axis.In the equation \(-y^2 + y + 1 = 0\), which is rearranged to \(-1y^2 + 1y + 1 = 0\), the coefficients are \( a = -1 \), \( b = 1 \), and \( c = 1 \). These values are substituted into the quadratic formula to find the solution.

Solving Quadratic Equations

Solving a quadratic equation involves finding the values of \( y \) that make the equation equal to zero. Here’s a simple guide using the quadratic formula:1. **Identify the coefficients**: Rearrange the equation into the standard form \( ay^2 + by + c = 0 \). For example, in the equation \(-y^2 + y + 1 = 0\), \( a = -1 \), \( b = 1 \), and \( c = 1 \).2. **Apply the quadratic formula**: Substitute the identified coefficients into:\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]3. **Calculate the discriminant**: Evaluate \( b^2 - 4ac \) to understand the nature of the roots. If it’s positive, proceed to find two real roots.4. **Find the roots**: Solve for \( y \) by conducting the operations of addition and subtraction with the square root term separately.By following these steps, you'll be able to efficiently find the solution to any quadratic equation.