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Chapter 9: Problem 40

\(4 v^{2}+5 v-12=0\)

Short Answer

Expert verified
\(v = \frac{-5 \pm \sqrt{217}}{8}\)

Step by step solution

01

Identify the coefficients

The given equation is in the form of a quadratic equation: \(4v^2 + 5v - 12 = 0\). Identify the coefficients: \(a = 4\), \(b = 5\), and \(c = -12\).
02

Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \(D = b^2 - 4ac\). Substitute the values of \(a, b,\) and \(c\): \(D = 5^2 - 4(4)(-12) = 25 + 192 = 217\).
03

Apply the quadratic formula

The solutions for the quadratic equation can be found using the quadratic formula: \(v = \frac{-b \pm \sqrt{D}}{2a}\). Since \(D = 217\), \(a = 4\), and \(b = 5\), substitute these values into the formula: \(v = \frac{-5 \pm \sqrt{217}}{8}\).
04

Simplify the solutions

Write the two solutions explicitly: \(v_1 = \frac{-5 + \sqrt{217}}{8}\) and \(v_2 = \frac{-5 - \sqrt{217}}{8}\).

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Key Concepts

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

Coefficients
In a quadratic equation, the coefficients are the numerical values that multiply the variables. A standard quadratic equation is written in the form: \(ax^2 + bx + c = 0\). Here,
  • \(a\) is the coefficient of \(x^2\)
  • \(b\) is the coefficient of \(x\)
  • \(c\) is the constant term
For the equation in the exercise, \(4v^2 + 5v - 12 = 0\), we identify the coefficients as follows:
  • \(a = 4\)
  • \(b = 5\)
  • \(c = -12\)
Understanding the coefficients is crucial because they are used in further steps to find the solutions.
Discriminant
The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) helps us determine the nature of the roots without actually solving the equation. It's calculated using the formula: \(D = b^2 - 4ac\)For our given equation with coefficients \(a = 4\), \(b = 5\), and \(c = -12\), the discriminant is calculated as: \[D = 5^2 - 4(4)(-12) = 25 + 192 = 217\]The value of the discriminant can tell us:
  • If \(D > 0\), there are two distinct real roots.
  • If \(D = 0\), there is one repeated real root.
  • If \(D < 0\), there are no real roots, but two complex roots.
Here, \(D = 217 > 0\), so our equation has two distinct real roots.
Quadratic Formula
The quadratic formula is a way to find the solutions of a quadratic equation \(ax^2 + bx + c = 0\). It is derived from completing the square of the quadratic equation. The formula is: \[x = \frac{-b \pm \sqrt{D}}{2a}\]Where:
  • \(D\) is the discriminant
  • \(a\), \(b\), and \(c\) are the coefficients
In our exercise, the quadratic formula helps us find the roots of \(4v^2 + 5v - 12 = 0\). Plugging in the values \(a = 4\), \(b = 5\), and \(D = 217\), we get: \[v = \frac{-5 \pm \sqrt{217}}{8}\]This gives us the solutions in the next step.
Solving Quadratic Equations
Solving a quadratic equation means finding the values of the variable that make the equation true. Using the quadratic formula, we have: \[v = \frac{-5 \pm \sqrt{217}}{8}\]This can be split into two separate solutions:
  • \(v_1 = \frac{-5 + \sqrt{217}}{8}\)
  • \(v_2 = \frac{-5 - \sqrt{217}}{8}\)
These two values are the solutions of the quadratic equation \(4v^2 + 5v - 12 = 0\). Make sure to substitute these back into the original equation to verify their correctness.

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Most popular questions from this chapter

\(2 n^{2}+6=-20\)

\(\begin{aligned} &\text { Problem: Simplify: } \sqrt{-18}\\\ &\text { Incorrect Answer: } \sqrt{-18}\\\ &=\sqrt{18} \sqrt{-1}\\\ &=\sqrt{18} i \end{aligned}\)

\(a^{2}+12 a+157=0\)

\(y=x^{2}-8 x+10\)

Problem: Use the quadratic formula to solve \(2 x^{2}-51 x+270=0\) Incorrect Answer: \(x=\frac{-51 \pm \sqrt{(-51)^{2}-4(2)(270)}}{2(2)}\) \(x=\frac{-51 \pm \sqrt{2601-2160}}{4}\) \(x=\frac{-51 \pm \sqrt{441}}{4}\) \(x=\frac{-51 \pm 21}{4}\) \begin{array}{lll} x=\frac{-51+21}{4} & \text { or } & x=\frac{-51-21}{4} \\ x=\frac{-30}{4} & \text { or } & x=\frac{-72}{4} \\ x=-\frac{15}{2} & \text { or } & x=-18 \end{array}
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