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Chapter 6: Problem 64

Solve the equations in Exercises 53-72 using the quadratic formula. \(9 x^{2}-12 x-5=0\)

Short Answer

Expert verified
The solutions to the quadratic equation \(9x^{2} - 12x - 5 = 0\) are \(x = 1\) and \(x = -5/9\)

Step by step solution

01

Identify the coefficients

The coefficients of the quadratic equation \(9 x^{2}-12 x-5=0\) are \(a = 9\), \(b = -12\), and \(c = -5\)
02

Substitute into the quadratic formula

Substitute the identified values \(a\), \(b\), \(c\) into the quadratic formula to get \(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4*9*(-5)}}{2*9}\)
03

Calculate the discriminant

Evaluate the expression inside the square root, namely, \(b^2 - 4ac\), to get \(144 - (-180) = 324\)
04

Solve for x

Substitute the calculated discriminant back into the quadratic formula to get \(x = \frac{12 \pm \sqrt{324}}{18}\), hence yielding the solutions \(x = 1\) or \(x = -5/9\)

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Key Concepts

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

Coefficients Identification
When solving a quadratic equation using the quadratic formula, the initial step involves identifying the coefficients of the equation. A quadratic equation is generally written in the standard form:
  • \(ax^2 + bx + c = 0\)
where \(a\), \(b\), and \(c\) are coefficients. In the given equation, \(9x^2 - 12x - 5 = 0\), these coefficients are:
  • \(a = 9\) (the coefficient of \(x^2\))
  • \(b = -12\) (the coefficient of \(x\))
  • \(c = -5\) (the constant term)
Identifying these coefficients correctly is crucial because they are directly used in the quadratic formula. The quadratic formula itself is:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Make sure to check the signs of the coefficients, as they can affect the calculations significantly. This precise identification sets a solid foundation to work through the quadratic formula in subsequent steps.
Discriminant Calculation
Once the coefficients are identified, the next step is calculating the discriminant. The discriminant is a key part of solving quadratic equations using the quadratic formula.
  • The discriminant is represented by the term \(b^2 - 4ac\) in the quadratic formula.
Calculating this helps us understand the nature of the roots of the equation:
  • If the discriminant is positive, the quadratic equation has two real and distinct solutions.
  • If the discriminant equals zero, there is exactly one real solution (or a repeated root).
  • If the discriminant is negative, the equation has two complex (imaginary) solutions.
For the equation \(9x^2 - 12x - 5 = 0\), substituting the values of \(a\), \(b\), and \(c\) into \(b^2 - 4ac\), we calculate:
  • \((-12)^2 - 4 \times 9 \times (-5) = 144 + 180 = 324\)
This positive discriminant indicates that the equation will have two real and distinct solutions. Understanding the role of the discriminant is vital in predicting the nature of solutions before proceeding to solve for \(x\).
Quadratic Equation Solution
After the discriminant is calculated and verified, we move on to finding the solutions of the quadratic equation using the quadratic formula. Recall the formula:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
For the quadratic equation \(9x^2 - 12x - 5 = 0\), substitute the coefficients and the discriminant back into this formula:
  • \(x = \frac{-(-12) \pm \sqrt{324}}{2 \times 9}\)
Simplifying this:
  • \(x = \frac{12 \pm 18}{18}\)
This yields two potential solutions:
  • \(x = \frac{12 + 18}{18} = 1\)
  • \(x = \frac{12 - 18}{18} = -\frac{5}{9}\)
Thus, the solutions to the equation are \(x = 1\) and \(x = -\frac{5}{9}\). Utilizing the quadratic formula not only provides specific solutions but also demonstrates a systematic approach to solving quadratic equations, contributing to a deeper understanding of polynomials.

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Most popular questions from this chapter

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(x^{2}-4 x-5\)

Solve each equation using the zero-product principle. \((x+11)(x-5)=0\)

Solve the equations in Exercises 53-72 using the quadratic formula. \(x^{2}+4 x=6\)

A phone plan costs \(\$ 20\) per month for 60 calling minutes Additional time costs \(\$ 0.40\) per minute. The formula $$ C=20+0.40(x-60) $$ gives the monthly cost for this plan, \(C\), for \(x\) calling minutes, where \(x>60\). How many calling minutes are possible for a monthly cost of at least \(\$ 28\) and at most \(\$ 40\) ?

Solve the quadratic equations in Exercises 37-52 by factoring. \(x^{2}-12 x=-36\)
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