Solve X^2+x-10=0: Quadratic Formula Guide

Hey guys! Let's dive into solving the quadratic equation $x^2 + x - 10 = 0$ using the quadratic formula. It might sound intimidating, but trust me, it's a straightforward process once you get the hang of it. We're going to break it down step by step so you can tackle any quadratic equation that comes your way. So, grab your pencils, and let's get started!

Understanding the Quadratic Formula

Before we jump into solving our specific equation, let's quickly review the quadratic formula itself. The quadratic formula is a powerful tool used to find the solutions (also called roots or zeros) of any quadratic equation in the standard form of $ax^2 + bx + c = 0$. The formula is given by:

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

Where:

• a is the coefficient of the $x^2$ term.
• b is the coefficient of the $x$ term.
• c is the constant term.

The $\pm$ symbol means we'll have two possible solutions, one where we add the square root and one where we subtract it. This is because quadratic equations can have up to two distinct real roots.

The part under the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us a lot about the nature of the roots:

• If $b^2 - 4ac > 0$, the equation has two distinct real roots.
• If $b^2 - 4ac = 0$, the equation has exactly one real root (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex roots.

Understanding these basics is crucial because it sets the stage for correctly applying the formula and interpreting the results. Without a clear grasp of what each term represents and how the discriminant influences the nature of the roots, you might find yourself getting lost in the calculations. So, take a moment to ensure you're comfortable with these concepts before moving on. Got it? Great, let’s proceed!

Identifying a, b, and c

Alright, now that we've refreshed our understanding of the quadratic formula, let's identify the values of a, b, and c in our equation: $x^2 + x - 10 = 0$. This is a crucial step because plugging the wrong values into the formula will obviously lead to incorrect solutions. Remember, a is the coefficient of the $x^2$ term, b is the coefficient of the x term, and c is the constant term.

In our equation:

• The coefficient of $x^2$ is 1, so a = 1.
• The coefficient of x is 1, so b = 1.
• The constant term is -10, so c = -10.

Double-check these values to make sure you've identified them correctly. A common mistake is to miss the negative sign in front of the constant term, so pay close attention to that. Once you're confident with your a, b, and c values, you're ready to plug them into the quadratic formula.

This step might seem simple, but it's a foundational element in solving quadratic equations. Getting these values right from the beginning ensures that the rest of your calculations will be accurate, saving you from potential frustration down the line. So, take your time, double-check your work, and make sure you're solid on these values before moving forward. Trust me, it'll make the rest of the process much smoother!

Applying the Quadratic Formula

Okay, with a = 1, b = 1, and c = -10, we can now plug these values into the quadratic formula:

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

Substitute the values:

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

Now, let's simplify step by step. First, calculate the value inside the square root:

$1^2 - 4(1)(-10) = 1 + 40 = 41$

So our equation becomes:

$x = \frac{-1 \pm \sqrt{41}}{2}$

This gives us two possible solutions:

$x_1 = \frac{-1 + \sqrt{41}}{2}$

$x_2 = \frac{-1 - \sqrt{41}}{2}$

These are the exact solutions to the quadratic equation. If you need decimal approximations, you can use a calculator to find the square root of 41 (approximately 6.403) and then perform the calculations:

$x_1 ≈ \frac{-1 + 6.403}{2} ≈ \frac{5.403}{2} ≈ 2.7015$

$x_2 ≈ \frac{-1 - 6.403}{2} ≈ \frac{-7.403}{2} ≈ -3.7015$

So, the approximate solutions are $x_1 ≈ 2.7015$ and $x_2 ≈ -3.7015$.

Remember, the key here is to carefully substitute the values and then simplify step by step. Don't try to rush through the calculations, as that's where mistakes often happen. Take your time, double-check each step, and make sure you're following the order of operations correctly. With a little practice, you'll become a pro at applying the quadratic formula!

Verifying the Solutions

To ensure that our solutions are correct, we can plug them back into the original equation $x^2 + x - 10 = 0$ and see if they satisfy the equation. This step is crucial because it helps us catch any errors we might have made during the calculation process. It's like a final check to make sure everything adds up correctly.

Let's start with the first solution, $x_1 = \frac{-1 + \sqrt{41}}{2}$:

$\left(\frac{-1 + \sqrt{41}}{2}\right)^2 + \frac{-1 + \sqrt{41}}{2} - 10$

This looks a bit complicated, but let's break it down. Squaring the term and simplifying, we should get a value close to zero if the solution is correct. Without going through the full simplification here (which can be a bit tedious), we can rely on our earlier decimal approximation $x_1 ≈ 2.7015$. Plugging this in:

$(2.7015)^2 + 2.7015 - 10 ≈ 7.298 + 2.7015 - 10 ≈ 9.9995 - 10 ≈ -0.0005$

This is very close to zero, which indicates that our solution is likely correct. The slight difference is due to rounding errors in our decimal approximation.

Now let's check the second solution, $x_2 = \frac{-1 - \sqrt{41}}{2}$. Using the decimal approximation $x_2 ≈ -3.7015$:

$(-3.7015)^2 + (-3.7015) - 10 ≈ 13.691 - 3.7015 - 10 ≈ 9.9895 - 10 ≈ -0.0105$

Again, this is very close to zero, further confirming that our solutions are correct. The slight deviation is due to the decimal approximation. If we were to use the exact solutions, we would get exactly zero.

So, by plugging our solutions back into the original equation and verifying that they satisfy it (or come very close to satisfying it, given rounding errors), we can be confident that we have found the correct solutions to the quadratic equation.

Final Answer

Therefore, the solutions to the quadratic equation $x^2 + x - 10 = 0$ using the quadratic formula are:

$x_1 = \frac{-1 + \sqrt{41}}{2}, x_2 = \frac{-1 - \sqrt{41}}{2}$

Or, approximately:

$x_1 ≈ 2.7015, x_2 ≈ -3.7015$

So there you have it! We've successfully solved the quadratic equation using the quadratic formula. Remember, practice makes perfect, so keep working on these problems, and you'll become a quadratic equation-solving master in no time! Great job, guys!