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Chapter 16: Problem 34

Solve by using the quadratic formula. $$9 t^{2}=30 t+17$$

Short Answer

Expert verified
The solutions to the quadratic equation are \(t_1 = \frac{30 + \sqrt{288}}{18}\) and \(t_2 = \frac{30 - \sqrt{288}}{18}\)

Step by step solution

01

Identify the coefficients

In the quadratic equation \(9t^{2} - 30t + 17 = 0\), the coefficients are \(a = 9\), \(b = -30\), and \(c = 17\).
02

Calculate the discriminant

The discriminant \(D\) in the quadratic formula is given by \(D = b^{2}-4ac\). So, substitute the given coefficients into the equation to get: \(D = (-30)^{2} - 4*9*17 = 900 - 612 = 288\).
03

Apply the quadratic formula

The quadratic formula is given by \(t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). Use this formula along with the calculated discriminant to find the solution to the problem. Now, we鈥檒l substitute \(a = 9\), \(b = -30\), and \(D = 288\) into the quadratic formula which gives us two solutions: \(t_1 = \frac{30 + \sqrt{288}}{18}\) and \(t_2 = \frac{30 - \sqrt{288}}{18}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Quadratic Equations
Quadratic equations are a staple of algebra and come in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known values, and \( x \) represents the unknown variable that we wish to solve for. One powerful tool for solving these equations is the quadratic formula. It provides a straightforward solution for finding the roots of any quadratic equation.

The general steps for solving a quadratic using the quadratic formula include setting the equation to zero, identifying the coefficients \( a \), \( b \), and \( c \), computing the discriminant, and then applying the formula itself:
  • Set to Zero: Ensure that the quadratic equation is in the form \( ax^2 + bx + c = 0 \).
  • Identify Coefficients: Recognize the coefficients of \( a \), \( b \), and \( c \) in your equation.
  • Compute Discriminant: Calculate the discriminant, \( D \), which helps determine the nature and number of solutions.
  • Apply Formula: Use the coefficients and discriminant in the quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \), and simplify to find the values of \( x \).
Discriminant in Quadratics
The discriminant in the context of quadratic equations is a specific expression denoted as \( D = b^2 - 4ac \). It plays a crucial role in determining not just the solutions, but also the nature of the solutions. There are three different scenarios:
  • Positive Discriminant: If the discriminant is positive, it means there are two distinct real solutions to the quadratic equation.
  • Zero Discriminant: If the discriminant is zero, the quadratic has exactly one real solution, sometimes referred to as a repeated or double root.
  • Negative Discriminant: A negative discriminant indicates that the equation has complex solutions, which means they have a real part and an imaginary part.
In our given exercise, the computed discriminant is 288, which is positive. This confirms that there are two different real solutions, given by the two possible values obtained when applying \( \pm \) in the quadratic formula.
Identifying Coefficients
In the realm of quadratic equations, identifying the correct coefficients is essential for applying the quadratic formula accurately. The coefficients of a quadratic equation \( ax^2 + bx + c = 0 \) are as follows:
  • \( a \): The coefficient of \( x^2 \), which determines the equation's leading term and influences its shape and orientation on a graph.
  • \( b \): The coefficient of \( x \), which affects the equation's symmetry and the location of its vertex along the x-axis.
  • \( c \): The constant term, which determines where the parabola intersects the y-axis.
Once you have accurately identified these coefficients, as we have done with \( a = 9 \), \( b = -30 \), and \( c = 17 \), it sets the stage for calculating the discriminant and ultimately applying the quadratic formula to find the solution to the equation.

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