Problem 49
Question
Beginning with the general form of a quadratic equation, \(a x^{2}+b x+c=0,\) solve for \(x\) by using the completing the square method, thus deriving the quadratic formula.
Step-by-Step Solution
Verified Answer
The quadratic formula derived is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
1Step 1: Start with the General Equation
The general form of a quadratic equation is given by:\[ a x^2 + b x + c = 0 \]
2Step 2: Divide Through by 'a'
Since completing the square is easier if the coefficient of \(x^2\) is 1, divide the entire equation by \(a\):\[ x^2 + \frac{b}{a} x + \frac{c}{a} = 0 \]
3Step 3: Move the Constant Term to the Other Side
Rearrange the equation to isolate the constant term on one side:\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \]
4Step 4: Complete the Square
To complete the square, add and subtract the square of half the coefficient of \(x\). Half of \(\frac{b}{a}\) is \(\frac{b}{2a}\), and its square is \(\frac{b^2}{4a^2}\). Add and subtract this value:\[ x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} \]
5Step 5: Rewrite the Left Side as a Square
The left side of the equation can now be written as a perfect square:\[ \left( x + \frac{b}{2a} \right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \]
6Step 6: Simplify the Right Side
Combine the terms on the right-hand side over a common denominator:\[ \left( x + \frac{b}{2a} \right)^2 = \frac{-4ac + b^2}{4a^2} \]
7Step 7: Take Square Roots of Both Sides
Take the square root of both sides, remembering to consider both the positive and negative roots:\[ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \]
8Step 8: Solve for 'x'
Isolate \(x\) by subtracting \(\frac{b}{2a}\) from both sides:\[ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \]Combine the terms over the common denominator:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Key Concepts
Quadratic EquationQuadratic FormulaStep-by-Step SolutionPerfect Square
Quadratic Equation
A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. This type of equation can model various real-world situations, such as predicting the height of a thrown ball over time.
Quadratic equations are characterized by the highest power of the variable \( x \) being squared (\( x^2 \)).To solve a quadratic equation means finding the values of \( x \) that make the equation true. These solutions are also known as the roots of the equation. Depending on the specific values of \( a \), \( b \), and \( c \), a quadratic equation may have:
Quadratic equations are characterized by the highest power of the variable \( x \) being squared (\( x^2 \)).To solve a quadratic equation means finding the values of \( x \) that make the equation true. These solutions are also known as the roots of the equation. Depending on the specific values of \( a \), \( b \), and \( c \), a quadratic equation may have:
- Two distinct real roots
- One real root (a repeated root)
- No real roots, but two complex roots
Quadratic Formula
The quadratic formula provides a straightforward way to find the roots of any quadratic equation. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula originates from the process of completing the square, as demonstrated in the exercise.
Here's how each part of the quadratic formula functions:
Here's how each part of the quadratic formula functions:
- **\(-b\)**: The formula starts by negating the coefficient of \( x \), which is \( b \).
- **\(\pm\sqrt{b^2 - 4ac}\)**: This part considers both the positive and negative square roots, which means there could be two different solutions.
- **The Discriminant \((b^2 - 4ac)\)**: Determines the nature of the roots. If the discriminant is positive, there are two real roots. If it is zero, there is exactly one real root. If it is negative, the roots are complex.
- **Divided by \(2a\)**: Completes the formula by accounting for the initial coefficient of \( x^2 \).
Step-by-Step Solution
Solving a quadratic equation using the completing the square method involves a series of structured steps that transform the expression into a form that reveals its roots. Here's a recap and insights into each step:
Start with the General Equation: Begin with \( ax^2 + bx + c = 0 \). This lays out the basic form to adjust.
Divide Through by 'a': Simplifying by dividing the equation by \( a \) makes the coefficient of \( x^2 \) equal to 1, easing further manipulation: \[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \]
Start with the General Equation: Begin with \( ax^2 + bx + c = 0 \). This lays out the basic form to adjust.
Divide Through by 'a': Simplifying by dividing the equation by \( a \) makes the coefficient of \( x^2 \) equal to 1, easing further manipulation: \[ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \]
Move the Constant Term: Shifting constant terms to the other side prepares the quadratic term for modification: \[ x^2 + \frac{b}{a}x = -\frac{c}{a} \]
Complete the Square: Find the perfect square trinomial by adding and subtracting the square of half the \( x \)-term coefficient. This is where the equation reshapes itself heavily: \[ x^2 + \frac{b}{a}x + \frac{\left(\frac{b}{2a}\right)^2} = -\frac{c}{a} + \frac{\left(\frac{b^2}{4a^2}\right) \]
Transform the Expression: The left side is now a perfect square: \[ \left(x+\frac{b}{2a}\right)^2 \]
Simplify Further Steps: Tidy up the equation by performing algebraic operations.
Solver's Insight: Always remember to consider both the positive and negative roots when using the square root in this method, as indicated in Step 7 and Step 8. This solution method not only reveals the answers but fundamentally illustrates why the quadratic formula works.
Solver's Insight: Always remember to consider both the positive and negative roots when using the square root in this method, as indicated in Step 7 and Step 8. This solution method not only reveals the answers but fundamentally illustrates why the quadratic formula works.
Perfect Square
A perfect square in algebra refers to an expression that is the square of a binomial. This transformation is crucial in the method of completing the square. Understanding what a perfect square looks like is key:For example, \( (x + d)^2 = x^2 + 2dx + d^2 \) is a perfect square trinomial.
The process of finding a perfect square involves:
The process of finding a perfect square involves:
- Identifying the square trinomial form \( x^2 + 2dx + d^2 \)
- Determining "\( d \)" from half the coefficient of \( x \), i.e., for \( x^2 + bx \), \( d = \frac{b}{2} \)
- Realizing that by adjusting the equation to include \( d^2 \) you transform it into \( (x + d)^2 \) making solving the equation simpler
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Problem 49
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