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Chapter 9: Problem 34

Represent the solution graphically. Check the solution algebraically. $$ x^{2}+2 x=3 $$

Short Answer

Expert verified
The solutions of the equation are x = 1 and x = -3

Step by step solution

01

Rearrange the equation

First, simplify the equation by moving the constant term to one side. So the equation becomes, \(x^{2} + 2x - 3 = 0\)
02

Plot the function

The function corresponding to the equation is \(y = x^{2} + 2x - 3\). By plotting this function, you will get a parabola and can read its roots from the graph, which are the x-values where the function intersects the x-axis.
03

Solve for x using the quadratic formula

The quadratic formula is given by \(-\frac{b}{2a} \pm \frac{\sqrt{b^2-4ac}}{2a}\), where a, b and c are coefficients of the quadratic equation. In this case, a = 1, b = 2 and c = -3. Substituting these values into the formula, you find that the solutions are \(x = 1\) and \(x = -3\)
04

Check the solution algebraically

Substitute the solution back into the original equation to verify. For \(x = 1\), \(1^2 + 2*1 = 3\), which is true. For \(x = -3\), \((-3)^2 + 2*-3 = 3\), which is also true. So, the roots are correct

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Key Concepts

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

Parabola
A parabola is a U-shaped curve that appears in the graph of a quadratic equation like the one given in the exercise: \(y = x^2 + 2x - 3\).
Whenever you plot a quadratic equation, you will see this characteristic shape. The highest or the lowest point of a parabola is called the "vertex." In this particular equation, the parabola opens upwards because the coefficient of \(x^2\) is positive.
Some important features of a parabola include:
  • Axis of Symmetry: A vertical line that divides the parabola into two mirror-image halves. In our equation, the axis of symmetry can be found using the formula \(x = -\frac{b}{2a}\).
  • Vertex: The turning point of the parabola, found at the axis of symmetry. For our function, it's located at \((-1, -4)\).
  • Intercepts: The points where the parabola cuts the axes. The x-intercepts are the solutions to the equation, which we'll find using the quadratic formula. There's also a y-intercept at \(y = -3\), where the parabola crosses the y-axis.
Quadratic Formula
The quadratic formula is a universal tool for solving quadratic equations like \(x^2 + 2x - 3 = 0\).
When factoring doesn't work or isn't obvious, the quadratic formula comes to the rescue. It is given by:
\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]where \(a\), \(b\), and \(c\) are the coefficients in the quadratic equation \(ax^2 + bx + c = 0\).
  • In our exercise, identify the coefficients: \(a = 1\), \(b = 2\), and \(c = -3\).
  • Substitute these values into the formula to find the roots of the equation.
  • This formula efficiently finds real or complex roots depending on the discriminant \(b^2 - 4ac\). If the discriminant is positive, you get two real roots. If it's zero, there is one real root. A negative discriminant indicates complex roots.
For our equation, the solutions found are \(x = 1\) and \(x = -3\), as these make the discriminant \(2^2 - 4 \cdot 1 \cdot (-3) = 16\) positive.
Roots of Equations
Finding the roots of a quadratic equation means determining the values of \(x\) that make the equation true鈥攅ssentially, where it touches or crosses the x-axis in the graph.
These x-values are also known as the "solutions" or "zeros" of the equation. In our quadratic equation \(x^2 + 2x - 3 = 0\), we found the roots by plotting the parabola and using the quadratic formula.
  • Double-checking with a graph: Graphing the equation and observing where the parabola intersects the x-axis can visually confirm the roots, which can be seen at \(x = 1\) and \(x = -3\).
  • Verification: Substitute the roots back into the original equation to ensure the solution satisfies it. If substituting \(x = 1\) and \(x = -3\) results in a true equation, the solutions are correctly identified.
  • Real-life applications: Understanding roots is essential for solving real-world problems, such as physics equations describing motion, economics models, or any scenario where predicting outcomes is important.
Remember, the roots give crucial information about the behavior of quadratic relationships.

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Most popular questions from this chapter

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$7 x^{2}-63=0$$

Writing Explain how you can use this two-part form of the quadratic formula $$x=\frac{-b}{2 a} \pm \frac{\sqrt{b^{2}-4 a c}}{2 a}$$ to find the distance between the axis of symmetry of a parabola and either of its \(x\) -intercepts.

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$3 x^{2}=6$$

A boulder falls off the top of a cliff during a storm. The cliff is 60 feet high. Find how long it will take for the boulder to hit the road below. Which problem solving method do you prefer? Why?

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is negative, then the equation has no real solution.
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