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Chapter 6: Problem 59

Solve the equations using the quadratic formula. \(x^{2}+4 x-7=0\)

Short Answer

Expert verified
The solutions to the quadratic equation are \(x = -2 + \sqrt{11}\) and \(x = -2 - \sqrt{11}\)

Step by step solution

01

Identify the coefficients

In the given equation \(x^{2}+4x-7=0\), the coefficients are \(a=1\), \(b=4\), and \(c=-7\). These will be plugged into the quadratic formula.
02

Calculate the discriminant

The discriminant is the part inside the square root in the quadratic formula, i.e., \(b^{2} - 4ac\). We have \(b^{2} - 4ac = 4^{2} - 4*1*(-7) = 16 + 28 = 44\).
03

Apply the quadratic formula

Use the values in the quadratic formula to determine x: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} = \frac{-4 \pm \sqrt{44}}{2*1}\]. This simplifies to \[x = -2 \pm \sqrt{11}\].

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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 mathematical expressions that take the form \( ax^2 + bx + c = 0 \). In this expression, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). Quadratic equations are crucial in mathematics because they form the basis for many problem-solving scenarios, enabling a wide array of applications from physics to finance.

Here is a breakdown of each component:
  • **\( a \)**: The coefficient of \( x^2 \), which determines the parabola's width and direction (upwards if positive, downwards if negative).
  • **\( b \)**: The coefficient of \( x \), which affects the position of the parabola along the x-axis.
  • **\( c \)**: The constant term, which moves the parabola up or down along the y-axis.
These equations often have two solutions, representing the points where the parabola intersects the x-axis. The solutions can be found using various methods, with the quadratic formula being one of the most reliable.
Discriminant Calculation
The discriminant is an essential component when working with the quadratic formula. It is given by \( b^2 - 4ac \) and determines the nature and number of the roots of the quadratic equation. Here鈥檚 how it affects the solutions:

  • If the discriminant is **positive**, the equation has two distinct real roots.
  • If the discriminant is **zero**, there is exactly one real root (or a repeated root).
  • If the discriminant is **negative**, the roots are complex or imaginary, meaning they do not intersect the real number line.
Given the equation \(x^2 + 4x - 7 = 0\), the discriminant is calculated as follows:\[b^2 - 4ac = 4^2 - 4(1)(-7) = 16 + 28 = 44\]Since \(44\) is positive, this equation has two distinct real roots.
Mathematical Problem Solving
Problem-solving with quadratic equations involves identifying the best approach to use based on the given data. In our example, the quadratic formula is recommended due to its universal applicability. Let's go through its application:
  • **Step 1**: Identify the coefficients from the equation \( x^2 + 4x - 7 = 0 \). Here, \( a = 1 \), \( b = 4 \), \( c = -7 \).
  • **Step 2**: Calculate the discriminant using \( b^2 - 4ac \), to gauge the nature of the roots.
  • **Step 3**: Apply the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].Using our identified values, this yields:\[x = \frac{-4 \pm \sqrt{44}}{2}\]
  • **Step 4**: Simplify to get the final roots: \( x = -2 \pm \sqrt{11} \).
The steps ensure that you process the equation correctly, leading to the accurate and reliable determination of the solutions. This method of solving helps maintain clarity and precision, advantageous both in academic contexts and real-world applications.

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Most popular questions from this chapter

Solve each equation by the method of your choice. \(\frac{x^{2}}{3}-\frac{x}{2}-\frac{3}{2}=0\)

Use FOIL to find the products. \((2 x-9)(7 x-4)\)

Solve the equations using the quadratic formula. \(x^{2}+4 x+1=0\)

Factor the trinomials , or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(3 x^{2}-x-2\)

Solve the equations using the quadratic formula. \(x^{2}-3 x=10\)
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