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Chapter 8: Problem 49

Find the discriminant. Use it to determine whether the solutions for each equation are A. two rational numbers B. one rational number C. two irrational numbers D. two nonreal complex numbers. Tell whether the equation can be solved using the zero-factor property, or if the quadratic formula should be used instead. Do not actually solve. $$ x^{2}+4 x+2=0 $$

Short Answer

Expert verified
Two irrational numbers. Use the quadratic formula.

Step by step solution

01

- Identify coefficients

In the quadratic equation given by \(ax^2 + bx + c = 0\), the coefficients can be identified as follows: \(a = 1\), \(b = 4\), and \(c = 2\).
02

- Write the formula for the discriminant

The formula for the discriminant is \(b^2 - 4ac\).
03

- Substitute the coefficients into the discriminant formula

Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(4^2 - 4(1)(2)\).
04

- Simplify the expression

Calculate the discriminant: \(16 - 8 = 8\).
05

- Determine the nature of the roots

Since the discriminant (8) is positive and not a perfect square, the quadratic equation has two irrational solutions.
06

- Decide the solving method

The zero-factor property cannot be used directly here, so the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) should be used.

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Key Concepts

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

form and examples
***
importance
b2-4ac ,based on the coefficients A and C
solving methods
There are different methods for solving quadratic equations. The two main methods include:
    From C or A to calculate the discrimnant or secondly finding the a,b coefficient

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Most popular questions from this chapter

Solve each problem. When appropriate, round answers to the nearest tenth. An object is projected directly upward from the ground. After \(t\) seconds its distance in feet above the ground is $$ s(t)=144 t-16 t^{2} $$ After how many seconds will the object be \(128 \mathrm{ft}\) above the ground? (Hint: Look for a common factor before solving the equation.)

Solve each problem. When appropriate, round answers to the nearest tenth. Manuel is planting a vegetable garden in the shape of a right triangle. The longer leg is \(3 \mathrm{ft}\) longer than the shorter leg, and the hypotenuse is \(3 \mathrm{ft}\) longer than the longer leg. Find the lengths of the three sides of the garden.

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In the 1939 classic movie The Wizard of Oz, Ray Bolger's character, the Scarecrow, wants a brain. When the Wizard grants him his "Th.D." (Doctor of Thinkology), the Scarecrow replies with the following statement. Scarecrow: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. His statement sounds like the formula for the Pythagorean theorem. To what kind of triangle does the Scarecrow refer in his statement? To what kind of triangle does the Pythagorean theorem actually refer?

The following exercises are not grouped by type. Solve each equation. $$\left(x-\frac{1}{2}\right)^{2}+5\left(x-\frac{1}{2}\right)-4=0$$
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