In The Derivation Of The Quadratic Formula By Completing The Square, The Equation $\[ \left(x+\frac{b}{2a}\right)^2=\frac{-4ac+b^2}{4a^2} \\]is Created By Forming A Perfect Square Trinomial.What Is The Result Of Applying The Square Root

Introduction

The quadratic formula is a fundamental concept in mathematics, used to solve quadratic equations of the form $ax^2 + bx + c = 0$. One of the methods used to derive the quadratic formula is by completing the square. This method involves manipulating the quadratic equation to form a perfect square trinomial, which can then be solved using the square root. In this article, we will explore the derivation of the quadratic formula by completing the square and examine the result of applying the square root.

Completing the Square

The process of completing the square involves manipulating the quadratic equation to form a perfect square trinomial. This is done by adding and subtracting a constant term to the equation, which allows us to write the equation in the form of a perfect square trinomial. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

To complete the square, we start by adding and subtracting the square of half the coefficient of the $x$ term. This can be written as:

$$ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c$$

Next, we add and subtract the square of half the coefficient of the $x$ term, which is $\left(\frac{b}{2a}\right)^2$. This can be written as:

$$a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2\right) + c - a\left(\frac{b}{2a}\right)^2$$

The expression inside the parentheses is a perfect square trinomial, which can be written as:

$$\left(x + \frac{b}{2a}\right)^2$$

Therefore, the quadratic equation can be written as:

$$a\left(x + \frac{b}{2a}\right)^2 = c - a\left(\frac{b}{2a}\right)^2$$

Derivation of the Quadratic Formula

To derive the quadratic formula, we need to isolate the square root term. We can do this by dividing both sides of the equation by $a$:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{c - a\left(\frac{b}{2a}\right)^2}{a}$$

Simplifying the right-hand side of the equation, we get:

$$\left(x + \frac{b}{2a}\right)^2 = \frac{-4ac+b^2}{4a^2}$$

Taking the square root of both sides of the equation, we get:

$$x + \frac{b}{2a} = \pm\sqrt{\frac{-4ac+b^2}{4a^2}}$$

Simplifying the right-hand side of the equation, we get:

$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

This is the quadratic formula, which can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$.

Applying the Square Root

When we apply the square root to the equation, we get two possible solutions:

$$x = \frac{-b + \sqrt{b^2-4ac}}{2a}$$

$$x = \frac{-b - \sqrt{b^2-4ac}}{2a}$$

These two solutions are the roots of the quadratic equation.

Conclusion

In this article, we have explored the derivation of the quadratic formula by completing the square. We have seen how the quadratic equation can be manipulated to form a perfect square trinomial, which can then be solved using the square root. We have also examined the result of applying the square root, which gives us two possible solutions to the quadratic equation. The quadratic formula is a powerful tool for solving quadratic equations, and it has many applications in mathematics and science.

References

• [1] "Quadratic Formula" by Math Open Reference
• [2] "Completing the Square" by Purplemath
• [3] "Quadratic Equations" by Khan Academy

Further Reading

• [1] "Quadratic Equations and Functions" by Math Is Fun
• [2] "Quadratic Formula and Completing the Square" by IXL
• [3] "Quadratic Equations and Functions" by CK-12 Foundation
  Quadratic Formula Q&A: Frequently Asked Questions =====================================================

Introduction

The quadratic formula is a fundamental concept in mathematics, used to solve quadratic equations of the form $ax^2 + bx + c = 0$. In our previous article, we explored the derivation of the quadratic formula by completing the square. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$. It is given by:

$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. The formula will then give you two possible solutions for the quadratic equation.

Q: What is the difference between the two solutions?

A: The two solutions are the roots of the quadratic equation. They are the values of $x$ that satisfy the equation.

Q: How do I determine which solution is correct?

A: To determine which solution is correct, you need to plug both solutions back into the original equation and check if they satisfy the equation. If they do, then both solutions are correct. If they don't, then one of the solutions is incorrect.

Q: Can I use the quadratic formula to solve quadratic equations with complex roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. The formula will give you two complex solutions, which are the roots of the equation.

Q: Can I use the quadratic formula to solve quadratic equations with rational roots?

A: Yes, you can use the quadratic formula to solve quadratic equations with rational roots. The formula will give you two rational solutions, which are the roots of the equation.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not plugging in the correct values of $a$, $b$, and $c$
• Not simplifying the expression under the square root
• Not checking if the solutions satisfy the original equation

Q: Can I use the quadratic formula to solve quadratic equations with negative coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with negative coefficients. The formula will give you two solutions, which are the roots of the equation.

Q: Can I use the quadratic formula to solve quadratic equations with fractional coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with fractional coefficients. The formula will give you two solutions, which are the roots of the equation.

Conclusion

In this article, we have answered some of the most frequently asked questions about the quadratic formula. We have seen how the quadratic formula can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. We have also seen some common mistakes to avoid when using the quadratic formula. By understanding the quadratic formula and how to use it, you can solve a wide range of quadratic equations.

References

• [1] "Quadratic Formula" by Math Open Reference
• [2] "Completing the Square" by Purplemath
• [3] "Quadratic Equations" by Khan Academy

Further Reading

• [1] "Quadratic Equations and Functions" by Math Is Fun
• [2] "Quadratic Formula and Completing the Square" by IXL
• [3] "Quadratic Equations and Functions" by CK-12 Foundation