Problem 89 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. $$ 7 y^{2}-9 y-17=0 $$

Step-by-Step Solution

Answer

The roots of the equation \( 7 y^{2}-9 y-17=0 \) are \( y = \frac{{9 + \sqrt{557}}}{14},\frac{{9 - \sqrt{557}}}{14}\)

Step 1: Identify the coefficients

The equation is in the form \( ay^2 + by + c = 0 \). Here \( a = 7, b = -9, c = -17 \).

Step 2: Apply the quadratic formula

The quadratic formula is \( y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Substituting the values of \( a, b, c \), we get \( y = \frac{{-(-9) \pm \sqrt{{(-9)^2 - 4*7*(-17)}}}}{2*7}\)

Step 3: Simplify the equation

After simplifying, we get \( y = \frac{{9 \pm \sqrt{{81+476}}}}{14} = \frac{{9 \pm \sqrt{557}}}{14}\)

Step 4: Find the roots

So, the roots of the equation are \( y = \frac{{9 + \sqrt{557}}}{14},\frac{{9 - \sqrt{557}}}{14}\)

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra. They are polynomial equations of the second degree, which means they include variables raised to the power of two. Specifically, a quadratic equation takes the form:

• \( ay^2 + by + c = 0 \)

Here, "\( a \)," "\( b \)," and "\( c \)" are the coefficients, and "\( y \)" is the variable. The "\( a \)" coefficient must not be zero for it to remain a quadratic equation.
These equations often represent parabolic graphs. Parabolas are U-shaped curves that can open upwards or downwards, determined by the sign of the "\( a \)" coefficient. Quadratic equations can have:

• Two real roots.
• One real root.
• No real roots.

The type of roots they have is usually determined by the discriminant, "\( b^2 - 4ac \)."
Understanding quadratic equations is crucial because they are prevalent in numerous mathematical problems and real-life scenarios, such as physics and engineering.

Coefficients in Polynomials

In polynomial equations like quadratics, coefficients play a significant role. They are the numerical or constant part of the terms of a polynomial. In the equation \( 7y^2 - 9y - 17 = 0 \), the coefficients are:

• \( a = 7 \) for \( y^2 \)
• \( b = -9 \) for \( y \)
• \( c = -17 \) for the constant term

These coefficients affect the shape and position of the parabola on a graph. Each term's power indicates the degree of the polynomial, and the highest degree dictates the highest power term, which in this case, is \( y^2 \).
When solving quadratic equations, identifying these coefficients accurately is crucial. They are used in the quadratic formula to find the roots. The quadratic formula involves substituting these values to solve the equation efficiently. Without correctly identifying the coefficients, solving the equation correctly would be impossible.

Solving Quadratic Equations

The most reliable method for solving quadratic equations is using the quadratic formula. This formula is extremely handy because it provides a direct way to solve any quadratic equation, regardless of the coefficients. The quadratic formula is:

• \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

By plugging in the coefficients \( a \), \( b \), and \( c \) from the given equation, we calculate the values of \( y \) that satisfy the equation.
Here’s a breakdown of the process:

• Identify coefficients \( a \), \( b \), and \( c \).
• Calculate the discriminant \( b^2 - 4ac \). The sign of the discriminant determines the nature of the roots.
• Substitute these into the quadratic formula.
• Simplify to find the values of \( y \) (the roots).

For the example equation \( 7y^2 - 9y - 17 = 0 \), you substitute \( a = 7 \), \( b = -9 \), and \( c = -17 \). After calculation, you find two solutions for \( y \), which are the roots of the equation. This systematic approach ensures that any quadratic equation can be tackled confidently.