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Chapter 2: Problem 31

Find the sum and the product of the roots of each quadratic equation. $$4 y^{2}-28 y+9=0$$

Short Answer

Expert verified
Sum of the roots: 7. Product of the roots: \(\frac{9}{4}\).

Step by step solution

01

Identify Coefficients

For the quadratic equation \(4y^2 - 28y + 9 = 0\), identify the coefficients: \(a = 4\), \(b = -28\), and \(c = 9\).
02

Use Vieta's Formulas

According to Vieta's formulas, the sum of the roots for the quadratic equation \(ay^2 + by + c = 0\) is given by \(-\frac{b}{a}\). The product of the roots is given by \(\frac{c}{a}\).
03

Calculate the Sum of the Roots

Use the formula for the sum of the roots: -\(\frac{b}{a} = -\frac{-28}{4} = 7\). Thus, the sum of the roots is 7.
04

Calculate the Product of the Roots

Use the formula for the product of the roots: \(\frac{c}{a} = \frac{9}{4}\). Thus, the product of the roots is \(\frac{9}{4}\).

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Key Concepts

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

Vieta's Formulas
Vieta's formulas are a powerful tool in algebra for understanding the relationship between the coefficients of a polynomial and its roots. They are named after the French mathematician Fran莽ois Vi猫te. These formulas give us an easy way to find the sum and product of the roots of a quadratic equation without having to actually find the roots themselves.

For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta鈥檚 formulas state:

  • The sum of the roots, often denoted as \(x_1 + x_2\), is given by \(-\frac{b}{a}\).
  • The product of the roots, denoted \(x_1 \cdot x_2\), is \(\frac{c}{a}\).

These simple formulas allow us to quickly extract key information about the roots from the equation鈥檚 coefficients. This is particularly useful when solving problems where calculating the exact roots is unnecessary.
Roots of Quadratic
The roots of a quadratic equation are the values of the variable that satisfy the equation, meaning they make the equation equal to zero. A quadratic equation, typically given in the format \(ax^2 + bx + c = 0\), can have either two real roots, one real root, or two complex roots, depending on the discriminant value (\(b^2 - 4ac\)).

To find the roots directly, we often use the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Where:
  • \(x\) denotes the roots,
  • \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation,
  • The \(\pm\) symbol indicates there are typically two solutions.

This formula is derived from completing the square method and provides an exact solution for the roots of any quadratic equation, allowing us to understand its roots comprehensively.
Sum and Product of Roots
The sum and product of the roots of a quadratic equation form the crux of Vieta's formulas. Knowing these relations provides insight into the equation's behaviour and properties without actually solving for the roots.

For any quadratic equation of the form \(ax^2 + bx + c = 0\):

  • The sum of the roots is \(x_1 + x_2 = -\frac{b}{a}\). This indicates an inherent connection between the coefficients of the linear term and the leading coefficient.
  • The product of the roots is given by \(x_1 \cdot x_2 = \frac{c}{a}\). This connects the constant term and the leading coefficient.

In practical terms, for the quadratic equation \(4y^2 - 28y + 9 = 0\), using these formulas, we find:
  • The sum of the roots is \(7\).
  • The product of the roots is \(\frac{9}{4}\).

Understanding these relationships is particularly beneficial because it allows for verification of the correctness of calculated or theoretically determined roots.

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Most popular questions from this chapter

(a) Use a graph to estimate the solution set for each inequality. Zoom in far enough so that you can estimate the relevant endpoints to the nearest thousandth. (b) Exercises \(61-70\) can be solved algebraically using the techniques presented in this section. Carry out the algebra to obtain exact expressions for the endpoints that you estimated in part (a). Then use a calculator to check that your results are consistent with the previous estimates. $$x^{2}+x-4 \leq 0$$

Solve the inequalities. Suggestion: A calculator may be useful for approximating key numbers. $$\frac{x^{2}-1}{x^{2}+8 x+15} \geq 0$$

Solve for the indicated letter. $$-\frac{1}{2} g t^{2}+v_{0} t+h_{0}=0 ; \text { for } t$$

Use zoom-in techniques to estimate the roots of each equation to the nearest hundredih, as in Example 6 and (b) use algebraic techniques to determine an exact expression for each root, then evaluate the expression and round to four decimal places. Check to see that your answers are consistent with the graphical results obtained in part (a). $$\sqrt{2 x-1}-\sqrt{x-2}=1$$

Solve \((x-a)^{2}-(x-b)^{2}>(a-b)^{2} / 4,\) where \(a\) and \(b\) are constants and \(a>b\).
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