Using the Quadratic Formula to Verify the Solutions

Oct 13, 2025 by magicalceylontours.com 104 views

In algebra, the quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$ . The formula is given by:

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

In this article, we will use the quadratic formula to verify the given solutions for two quadratic equations.

a. Verifying the Solutions for $x^2 + 9x + 20 = 0$

The given solutions for the equation $x^2 + 9x + 20 = 0$ are $x = -4$ and $x = -5$ . To verify these solutions, we will substitute them into the original equation and check if the equation holds true.

Substituting $x = -4$ into the Equation

Let's substitute $x = -4$ into the equation $x^2 + 9x + 20 = 0$ .

(-4)^2 + 9(-4) + 20 = 16 - 36 + 20 = 0

As we can see, the equation holds true when $x = -4$ . This confirms that $x = -4$ is indeed a solution to the equation.

Substituting $x = -5$ into the Equation

Now, let's substitute $x = -5$ into the equation $x^2 + 9x + 20 = 0$ .

(-5)^2 + 9(-5) + 20 = 25 - 45 + 20 = 0

Again, the equation holds true when $x = -5$ . This confirms that $x = -5$ is indeed a solution to the equation.

Using the Quadratic Formula to Verify the Solutions

To verify the solutions using the quadratic formula, we need to find the values of $a$ , $b$ , and $c$ in the equation $x^2 + 9x + 20 = 0$ . In this case, $a = 1$ , $b = 9$ , and $c = 20$ .

Now, let's plug these values into the quadratic formula:

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

x = \frac{-9 \pm \sqrt{9^2 - 4(1)(20)}}{2(1)}

x = \frac{-9 \pm \sqrt{81 - 80}}{2}

x = \frac{-9 \pm \sqrt{1}}{2}

x = \frac{-9 \pm 1}{2}

Simplifying the expression, we get:

x = \frac{-9 + 1}{2} \text{ or } x = \frac{-9 - 1}{2}

x = -4 \text{ or } x = -5

As we can see, the quadratic formula gives us the same solutions as the given solutions. This confirms that the given solutions are indeed correct.

b. Verifying the Solutions for $x^2 - 10x + 21 = 0$

The given solutions for the equation $x^2 - 10x + 21 = 0$ are $x = 3$ and $x = 7$ . To verify these solutions, we will substitute them into the original equation and check if the equation holds true.

Substituting $x = 3$ into the Equation

Let's substitute $x = 3$ into the equation $x^2 - 10x + 21 = 0$ .

(3)^2 - 10(3) + 21 = 9 - 30 + 21 = 0

As we can see, the equation holds true when $x = 3$ . This confirms that $x = 3$ is indeed a solution to the equation.

Substituting $x = 7$ into the Equation

Now, let's substitute $x = 7$ into the equation $x^2 - 10x + 21 = 0$ .

(7)^2 - 10(7) + 21 = 49 - 70 + 21 = 0

Again, the equation holds true when $x = 7$ . This confirms that $x = 7$ is indeed a solution to the equation.

Using the Quadratic Formula to Verify the Solutions

To verify the solutions using the quadratic formula, we need to find the values of $a$ , $b$ , and $c$ in the equation $x^2 - 10x + 21 = 0$ . In this case, $a = 1$ , $b = -10$ , and $c = 21$ .

Now, let's plug these values into the quadratic formula:

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

x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(21)}}{2(1)}

x = \frac{10 \pm \sqrt{100 - 84}}{2}

x = \frac{10 \pm \sqrt{16}}{2}

x = \frac{10 \pm 4}{2}

Simplifying the expression, we get:

x = \frac{10 + 4}{2} \text{ or } x = \frac{10 - 4}{2}

x = 7 \text{ or } x = 3

As we can see, the quadratic formula gives us the same solutions as the given solutions. This confirms that the given solutions are indeed correct.

Conclusion

In the previous article, we used the quadratic formula to verify the given solutions for two quadratic equations. In this article, we will answer some frequently asked questions about the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations of the form $ax^2 + bx + c = 0$ . The formula 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 identify the values of $a$ , $b$ , and $c$ in the quadratic equation. Then, you plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

Identify the values of $a$ , $b$ , and $c$ in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression to find the solutions.
Check if the solutions are real or complex.

Q: What is the difference between real and complex solutions?

A: Real solutions are solutions that can be expressed as a single number, such as $x = 3$ or $x = -4$ . Complex solutions are solutions that involve imaginary numbers, such as $x = 3 + 4i$ or $x = -4 - 3i$ .

Q: How do I determine if a solution is real or complex?

A: To determine if a solution is real or complex, you need to check if the expression under the square root is positive or negative. If the expression is positive, the solution is real. If the expression is negative, the solution is complex.

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 identifying the values of $a$ , $b$ , and $c$ correctly.
Not plugging the values into the quadratic formula correctly.
Not simplifying the expression correctly.
Not checking if the solutions are real or complex.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when simplifying the expression and checking if the solutions are real or complex.

Q: Are there any other ways to solve quadratic equations besides using the quadratic formula?

A: Yes, there are other ways to solve quadratic equations besides using the quadratic formula. Some of these methods include:

Factoring the quadratic equation.
Using the quadratic equation formula.
Graphing the quadratic equation.

Conclusion

In this article, we answered some frequently asked questions about the quadratic formula. We discussed how to use the quadratic formula, how to determine if a solution is real or complex, and some common mistakes to avoid. We also discussed other ways to solve quadratic equations besides using the quadratic formula.