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Chapter 10: Problem 2

Use the quadratic formula to solve each of the following quadratic equations. $$x^{2}+3 x-4=0$$

Short Answer

Expert verified
The solutions are \(x = 1\) and \(x = -4\).

Step by step solution

01

Identify Coefficients

First, identify the coefficients from the quadratic equation \(x^2 + 3x - 4 = 0\). Here, \(a = 1\), \(b = 3\), and \(c = -4\).
02

Write the Quadratic Formula

The quadratic formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). This formula will be used to find the roots of the quadratic equation.
03

Calculate the Discriminant

Calculate the discriminant using the formula \(b^2 - 4ac\). For this equation, it is \(3^2 - 4(1)(-4)\), which simplifies to \(9 + 16 = 25\).
04

Apply the Quadratic Formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula. With \(b = 3\) and discriminant \(= 25\), the formula becomes \(x = \frac{{-3 \pm \sqrt{25}}}{2(1)}\).
05

Simplify the Expression

Calculate the square root of the discriminant (\(\sqrt{25} = 5\)), and substitute back to get two solutions: \(x = \frac{{-3 + 5}}{2}\) and \(x = \frac{{-3 - 5}}{2}\).
06

Find the Solutions

Simplify further to find the solutions \(x = 1\) and \(x = -4\).

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Key Concepts

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

Quadratic Equations
Quadratic equations are a fundamental concept in algebra, typically taking the form \(ax^2 + bx + c = 0\). These equations can model various real-world phenomena, such as projectile motion or optimization problems. The key components of a quadratic equation are:
  • \(a\): the quadratic coefficient, which multiplies the \(x^2\) term
  • \(b\): the linear coefficient, which multiplies the \(x\) term
  • \(c\): the constant term, independent of \(x\)
Quadratic equations can have real or complex solutions, known as the roots. There are multiple methods to solve these equations, including factoring, completing the square, and using the quadratic formula. Each method has its own set of advantages depending on the specific equation being solved.
Discriminant
The discriminant is a crucial part of the quadratic formula, given by \(b^2 - 4ac\). It provides insight into the types of roots a quadratic equation will have, without solving the equation:
  • If the discriminant is positive, the equation has two distinct real roots.
  • If it equals zero, there is exactly one real root, sometimes called a repeated or double root.
  • If the discriminant is negative, the equation has two complex roots, which are conjugates of each other.
For the equation \(x^2 + 3x - 4 = 0\), the discriminant is calculated as \(3^2 - 4 \times 1 \times (-4)\), resulting in \(25\). Since 25 is positive, this indicates that the equation has two distinct real roots.
Roots of a Quadratic Equation
The roots of a quadratic equation are the solutions to the equation, where the graph of the quadratic function intersects the x-axis. These roots can be found using the quadratic formula: \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\). Let's break down the process:
  • Plug in coefficients: Substitute the values of \(a\), \(b\), and \(c\) into the formula. For \(x^2 + 3x - 4 = 0\), \(a = 1\), \(b = 3\), \(c = -4\).
  • Calculate using the discriminant: As we calculated earlier, \(b^2 - 4ac = 25\). This positive discriminant implies two real solutions.
  • Apply the formula: Substitute these values to get \(x = \frac{{-3 \pm 5}}{2}\), producing two solutions from the positive and negative square root.
  • Compute solutions: Simplifying gives \(x = 1\) and \(x = -4\).
These values represent the points where the parabola represented by the quadratic equation \(x^2 + 3x - 4 = 0\) crosses the x-axis.

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Most popular questions from this chapter

A class trip was to cost $$\$ 3000$$. If there had been ten more students, it would have cost each student $$\$ 25$$ less. How many students took the trip?

Each of three consecutive even whole numbers is squared. The three results are added and the sum is 596 . Find the numbers.

Find two numbers such that their sum is 4 and their product is \(1 .\)

The sum of a number and its reciprocal is \(\frac{73}{24}\). Find the number.

Use the quadratic formula to solve each of the following quadratic equations. $$4 n^{2}+8 n-1=0$$
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