Solving 10x^2 - 9x - 9 = 0 Find Real Solutions

Hey guys! Let's dive into solving this quadratic equation. We've got:

$$10x^2 - 9x - 9 = 0$$

We need to find the real solutions for x, and we've got a few methods we can use. Let's explore them!

Methods to Solve Quadratic Equations

Before we jump into solving, let's quickly recap our toolkit for tackling quadratic equations. There are primarily three methods we can use:

1. Factoring: This method involves breaking down the quadratic expression into a product of two binomials. If we can factor the equation, we can easily find the solutions by setting each factor equal to zero.

2. Quadratic Formula: This is the trusty formula that always works, regardless of whether the equation is easily factorable or not. The quadratic formula is given by:

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

   where a, b, and c are the coefficients of the quadratic equation in the form ax² + bx + c = 0.

3. Completing the Square: This method involves manipulating the equation to create a perfect square trinomial on one side. While it's a powerful method, it can be a bit more involved than the other two.

For this particular equation, let's start by trying to factor it. If that doesn't work, we'll confidently turn to the quadratic formula.

Attempting to Factor the Quadratic Equation

Okay, so we have the equation:

$$10x^2 - 9x - 9 = 0$$

To factor this, we need to find two binomials that, when multiplied together, give us the original quadratic expression. This means we're looking for something in the form:

$$(Ax + B)(Cx + D) = 10x^2 - 9x - 9$$

where A, B, C, and D are constants. The key here is to find the right combination of numbers that satisfy the equation. Specifically, we need two numbers that multiply to give us (10 * -9 = -90) and add up to -9.

Let's list the factor pairs of -90:

• 1 and -90
• -1 and 90
• 2 and -45
• -2 and 45
• 3 and -30
• -3 and 30
• 5 and -18
• -5 and 18
• 6 and -15
• -6 and 15
• 9 and -10
• -9 and 10

Looking at these pairs, we see that -15 and 6 add up to -9. Bingo! Now we can rewrite our middle term (-9x) using these numbers:

$$10x^2 - 15x + 6x - 9 = 0$$

Next, we factor by grouping. We group the first two terms and the last two terms:

$$(10x^2 - 15x) + (6x - 9) = 0$$

Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 5x, and from the second group, the GCF is 3:

$$5x(2x - 3) + 3(2x - 3) = 0$$

Notice that we now have a common factor of (2x - 3). We can factor this out:

$$(2x - 3)(5x + 3) = 0$$

We've successfully factored the quadratic equation! This is awesome because it makes finding the solutions super easy.

Finding the Solutions from Factored Form

Now that we have our equation factored as:

$$(2x - 3)(5x + 3) = 0$$

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

1. First factor:

   $$2x - 3 = 0$$

   Add 3 to both sides:

   $$2x = 3$$

   Divide by 2:

   $$x = \frac{3}{2}$$

2. Second factor:

   $$5x + 3 = 0$$

   Subtract 3 from both sides:

   $$5x = -3$$

   Divide by 5:

   $$x = -\frac{3}{5}$$

So, our solutions are x = 3/2 and x = -3/5. We found the real solutions!

Verifying the Solutions

It's always a good idea to double-check our answers to make sure we didn't make any mistakes. We can do this by plugging our solutions back into the original equation:

$$10x^2 - 9x - 9 = 0$$

1. Checking x = 3/2:

   $$10\left(\frac{3}{2}\right)^2 - 9\left(\frac{3}{2}\right) - 9 = 10\left(\frac{9}{4}\right) - \frac{27}{2} - 9 = \frac{90}{4} - \frac{27}{2} - 9 = \frac{45}{2} - \frac{27}{2} - \frac{18}{2} = \frac{45 - 27 - 18}{2} = \frac{0}{2} = 0$$

   It checks out!

2. Checking x = -3/5:

   $$10\left(-\frac{3}{5}\right)^2 - 9\left(-\frac{3}{5}\right) - 9 = 10\left(\frac{9}{25}\right) + \frac{27}{5} - 9 = \frac{90}{25} + \frac{27}{5} - 9 = \frac{18}{5} + \frac{27}{5} - \frac{45}{5} = \frac{18 + 27 - 45}{5} = \frac{0}{5} = 0$$

   This one checks out too!

We've confirmed that both solutions are correct.

Final Answer

The real solutions for the equation 10x² - 9x - 9 = 0 are:

$$x = \frac{3}{2} \quad \text{and} \quad x = -\frac{3}{5}$$

So, our solution set is {3/2, -3/5}.

Wrapping Up

We successfully solved the quadratic equation by factoring, which is often the quickest method when it works. We found the solutions x = 3/2 and x = -3/5 and verified them by plugging them back into the original equation. Remember, if factoring doesn't work, the quadratic formula is always a reliable backup!

Keep practicing, and you'll become a quadratic equation-solving pro in no time!