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Chapter 5: Problem 6

Solve each equation using the Quadratic Formula. $$ 2 x^{2}+3 x-5=0 $$

Short Answer

Expert verified
The solutions to the equation 2x^2 + 3x - 5 = 0 are x = 1 and x = -2.5.

Step by step solution

01

Identify the coefficients

In the quadratic equation of the form ax^2 + bx + c = 0, identify the values of a, b, and c. For the equation 2x^2 + 3x - 5 = 0, the coefficients are a = 2, b = 3, and c = -5.
02

Write down the Quadratic Formula

The Quadratic Formula to find the roots of a quadratic equation ax^2 + bx + c = 0 is given by x = (-b 卤 鈭�(b^2 - 4ac)) / (2a).
03

Plug the coefficients into the formula

Substitute the identified values into the Quadratic Formula: x = (-(3) 卤 鈭�((3)^2 - 4(2)(-5))) / (2(2)).
04

Simplify under the square root

First calculate the discriminant: (3)^2 - 4(2)(-5) = 9 + 40 = 49.
05

Calculate the values of x

Now, plug the discriminant back into the formula and simplify: x = (-3 卤 鈭�49) / 4. This gives two solutions: x = (-3 + 7) / 4 = 1 and x = (-3 - 7) / 4 = -2.5.

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Key Concepts

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

Quadratic Formula
The Quadratic Formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula provides a straightforward method for finding the roots of any quadratic equation. To apply the formula, you first need to identify the coefficients \( a \), \( b \), and \( c \) from the equation:

The General Quadratic Formula is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In this structure, the '卤' symbol indicates that the equation has two solutions, commonly referred to as roots. By substituting the coefficients into this formula, you can solve for \( x \) to find the roots of the quadratic equation.
Discriminant
The term under the square root in the Quadratic Formula, \( b^2 - 4ac \), is known as the discriminant. It is crucial in determining the nature and number of roots a quadratic equation will have. Based on the value of the discriminant:
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is exactly one real root (also called a repeated or double root).
  • If the discriminant is negative, there are no real roots, but two complex roots.
In our example \( 3^2 - 4 \cdot 2 \cdot (-5) \), the discriminant is 49, which is positive, suggesting our equation will have two distinct real roots.
Roots of Quadratic Equations
Roots of quadratic equations are the values of \( x \) that make the equation \( ax^2 + bx + c = 0 \) true. These are the solutions obtained after applying the Quadratic Formula. As demonstrated in our example, using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), you arrive at two values for \( x \). Plugging in the discriminant \( \sqrt{49} \) and coefficients into the equation gives us: \( x = \frac{-3 \pm 7}{4} \), resulting in the two solutions \( x = 1 \) and \( x = -2.5 \). These solutions represent the x-intercepts or the points where the parabola represented by the quadratic equation crosses the x-axis.
Simplifying Expressions
Simplifying expressions is a fundamental step in solving quadratic equations using the Quadratic Formula. It involves arithmetic operations such as addition, subtraction, multiplication, and finding square roots. To keep simplification manageable:
  • Always calculate the discriminant first.
  • Perform any necessary squares and multiplications before dealing with the subtraction or addition.
  • Use the calculated discriminant to simplify the square root term.
  • Finally, divide as needed to get the x values which are the roots of the equation.
In our provided equation, after calculating the discriminant (49), we then simplify the numerator by adding and subtracting 7 from -3, and finally divide by 4, giving us the simplified roots.

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Most popular questions from this chapter

Solve each equation using any method. When necessary, round real solutions to the nearest hundredth. For imaginary solutions, write exact solutions. $$ x^{2}-2 x+2=0 $$

What is the discriminant of a quadratic equation, and what does its value tell you about the solution(s) of the equation?

List the steps for solving the equation \(3 x^{2}-6=-7 x\) by the method of completing the square. Explain each step.

Simplify each expression. $$ (2-3 i)+(-4+5 i) $$

Solve each equation using the Quadratic Formula. Find the exact solutions. Then approximate any radical solutions. Round to the nearest hundredth. $$ 3 x^{2}+4 x-3=0 $$
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